Main

All-optical methods to characterize atomic quantum statistics were proposed for homogeneous Bose–Einstein condensates 9,10,11,12,13 and some modified spectral properties induced by Bose–Einstein condensates were attributed to collective emission9,10, recoil shifts12 or local field effects14.

We show a completely different phenomenon directly reflecting atom quantum statistics due to state-dependent dispersion. More precisely, the dispersion shift of a cavity mode depends on the atom number. If the atom number in some lattice region fluctuates from realization to realization, the modes get a fluctuating frequency shift. Thus, in the cavity transmission spectrum, resonances appear at different frequencies directly reflecting the atom number distribution function. Such a measurement allows us to calculate atomic statistical quantities, for example, mean value and variance reflected by spectral characteristics such as the central frequency and width.

Different phases of a degenerate gas possess similar mean-field densities but different quantum amplitudes. This leads to a superposition of different transmission spectra, which, for example, for a superfluid (SF) state consist of numerous peaks reflecting the discreteness of the matter field. Analogous discrete spectra reversing the role of atoms and light, thus reflecting the photon structure of electromagnetic fields, were obtained in cavity quantum electrodynamics (QED) with Rydberg atoms15 and solid-state superconducting circuits16. A quantum phase transition towards a Mott insulator (MI) state is characterized by a reduction of the number of peaks towards a single resonance, because atom number fluctuations are significantly suppressed17,18. As our detection scheme is based on non-resonant dispersive interaction independent of a particular level structure, it can also be applied to molecules19,20.

We consider the quantized motion of N two-level atoms in a deep periodic optical lattice with M sites formed by far off-resonance standing-wave laser beams1. A region of K≤M sites is coupled to two quantized light modes whose geometries (that is, axis directions or wavelengths) can be varied. This is shown in Fig. 1, which shows two cavities crossed by a one-dimensional (1D) string of atoms in equally separated wells generated by the lattice lasers (not shown). In practice, two different modes of the same cavity would do as well.

Figure 1: Schematic diagram of the set-up.
figure 1

Atoms are periodically trapped in an optical lattice created by laser beams, which are not shown. In addition, the atoms are illuminated by two light modes at angles θ0 and θ1 with respect to the lattice axis.

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As shown in the Methods section, the Heisenberg equations for the annihilation operators of two light modes al (l=0,1) with eigenfrequencies ωl and spatial mode functions ul(r) are

where l≠m, δl=g2/Δla, g is the atom–light coupling constant, Δla=ωl−ωa are the large cavity–atom detunings, κ is the cavity relaxation rate, ηl(t)=ηle−i ωl pt gives the external probe and are the atom number operators at a site with coordinate ri. We also introduce the operator of the atom number at illuminated sites .

In a classical limit, equation (1) corresponds to Maxwell’s equations with the dispersion-induced frequency shifts of cavity modes and the coupling coefficient between them . For a quantum gas, those quantities are operators, which will lead to striking results: atom number fluctuations will be directly reflected in such measurable frequency-dependent observables. Thus, cavity transmission spectra will reflect atomic statistics.

Equation (1) allows us to express the light operators al as a function of atomic occupation number operators and to calculate their expectation values for prescribed atomic states |Ψ〉. We start with the well-known examples of MI and SF states and generalize to any |Ψ〉 later.

From the viewpoint of light scattering, the MI state behaves almost classically as, for negligible tunnelling, precisely atoms are well localized at the ith site with no number fluctuations. It is represented by a product of Fock states, that is, , with expectation values

as . For simplicity, we consider equal average densities ().

In our second example, the SF state, each atom is delocalized over all sites leading to local number fluctuations at a lattice region with K<M sites. Mathematically it is a superposition of Fock states corresponding to all possible distributions of N atoms at M sites: . Although its average density is identical to an MI, it creates different light transmission spectra. Expectation values of light operators can be calculated from

representing a sum of all possible ‘classical’ terms. Thus, all these distributions contribute to scattering from an SF, which is obviously different from (2) with only a single contributing term.

In the simple case of only one mode a0 (a1≡0), the stationary solution of equation (1) for the photon number reads

where Δp=ω0p−ω0 is the probe–cavity detuning. We present calculated transmission spectra in Fig. 2 for the case where |u0(ri)|2=1 and reduces to . For a 1D lattice (see Fig. 1), this occurs for a travelling wave at any angle and a standing wave transverse (θ0=π/2) or parallel (θ0=0) to the lattice with atoms trapped at field maxima.

Figure 2: Photon number in a single-cavity mode.
figure 2

a, The single lorentzian for MI (red line) reflects the non-fluctuating atom number. Many lorentzians for SF (blue line) reflect atom number fluctuations, which are imprinted on the positions of narrow resonances. Here, κ is smaller than the satellite separation δ0 (κ=0.1δ0), N=M=30, K=15. b, The same as in a but κ=δ0 gives a smooth broadened contour for SF. c, Spectra for SF with N=M=70 and different number of sites illuminated K=10,35,68. The transmission spectra have different forms, as different atom distribution functions correspond to different K. κ=0.05δ0.

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For an MI, the averaging of equation (4) according to equation (2) gives the photon number 〈a0†a0〉MI, as a function of the detuning, as a single lorentzian described by equation (4) with width κ and frequency shift given by (equal to δ0NK in Fig. 2). Thus, for an MI, the spectrum reproduces a simple classical result of a lorentzian shifted owing to dispersion.

In contrast, for an SF, the averaging procedure of equation (3) gives a sum of lorentzians with different dispersion shifts corresponding to all atomic distributions |q1,…,qK〉. So, if each lorentzian is resolved, we can measure a comb-like structure by scanning the detuning Δp. In Fig. 2a,c, different shifts of the lorentzians correspond to different possible atom numbers at K sites (which owing to atom number fluctuations in SF, can take all values 0,1,2,…,N). The lorentzians are separated by δ0. Thus, we see that atom number fluctuations lead to the fluctuating mode shift, and hence to multiple resonances in the spectrum. For larger κ, the spectrum becomes continuous (Fig. 2b), but broader than that for MI.

Scattering of weak fields does not change the atom number distribution. However, as the SF is a superposition of different atom numbers in a region with K sites, a measurement projects the state into a subspace with fixed NK in this region, and a subsequent measurement on a timescale shorter than tunnelling between sites will yield the same result. The full spectrum of Fig. 2 is recovered by repeating the experiment or with sufficient delay to allow for redistribution via tunnelling. Such measurements will allow a time-dependent study of tunnelling and build-up of long-range order. Alternatively, measurements can be continued on the reduced subspace after changing a lattice region or light geometry.

We now consider two modes with ω0=ω1, the probe injected only into a0 (Fig. 1) and the mentioned geometries where (see Fig. 3). From equation (1), the stationary photon number is

where .

Figure 3: Photon number in one of two strongly coupled modes.
figure 3

a, Diffraction maximum, doublet for MI (red line) and spectrum with a structured right satellite for SF (blue line). Left satellites at zero detuning coincide for MI and SF. The structure in the satellite reflects atom number fluctuations in SF, whereas the narrow spectrum for MI demonstrates vanishing fluctuations. Here, κ is smaller than the satellite separation 2δ0 (κ=0.1δ0), K=15. b, The same as in a but κ=δ0 gives a broadened satellite for SF. c, Diffraction minimum, zero field for MI and structured spectrum for SF. The non-zero structured spectrum for SF reflects the fluctuating difference between atom numbers at odd and even sites, which exists even for the whole lattice illuminated, K=M. Here, κ is smaller than the satellite separation 2δ0 (κ=0.1δ0), K=30. d, The same as in c but κ=δ0 gives a broadened contour for SF. N=M=30 in all figures.

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In a classical (and the MI) case, equation (2) gives a two-satellite contour (5) reflecting normal-mode splitting of two oscillators 〈a0,1〉 coupled through atoms. This was recently observed21 for collective strong coupling, that is, the splitting exceeding κ. The splitting depends on the geometry (see equation (1)) representing diffraction of one mode into another. Thus, our results can be treated as scattering from a ‘quantum diffraction grating’ generalizing Bragg scattering, well known in different disciplines. In diffraction maxima (that is, u1*(ri)u0(ri)=1) we find providing the maximal classical splitting. In diffraction minima, we find providing both the classical splitting and photon number are almost zero.

In an SF, equation (3) shows that 〈a1†a1〉SF is given by a sum of all classical terms with all possible normal-mode splittings. In a diffraction maximum (Fig. 3a,b), the right satellite is split into components corresponding to all possible NK or extremely broadened. In a minimum (Fig. 3c,d), the splittings are determined by all differences between atom numbers at odd and even sites . Note that there is no classical description of the spectra in a minimum, because here the classical field (and 〈a1†a1〉MI) is simply zero for any Δp. Thus, for two cavities coupled at the diffraction minimum, the difference between the SF and MI states is even more striking: one has a structured spectrum instead of zero signal. Moreover, the difference between atom numbers at odd and even sites fluctuates even for the whole lattice illuminated, giving non-trivial spectra even for K=M.

In each of the examples in Figs 2 and 3, the photon number depends only on one statistical quantity, now called q, f(q1,…,qM)=f(q). For the single mode and two modes in a maximum, q is the atom number at K sites. For two modes in a minimum, q is the atom number at odd (or even) sites. Therefore, expectation values for some state |Ψ〉 can be reduced to , where pΨ(q) is the distribution function of q in this state.

In high-Q cavities (κ≪δ0=g2/Δ0a), f(q) is given by a narrow lorentzian of width κ peaked at some frequency proportional to q (q=0,1,…,N). The lorentzian height is independent of q. Thus, 〈f〉Ψ as a function of Δp represents a comb of lorentzians with the amplitudes simply proportional to pΨ(q).

This is our central result. It states that the transmission spectrum of a high-Q cavity 〈a†a(Δp)〉Ψ directly maps the distribution function of ultracold atoms pΨ(q), for example, the distribution function of atom number at K sites. Various atomic statistical quantities characterizing a particular state can be then calculated: mean value (given by the spectrum centre), variance (determined by the spectral width) and higher moments. Furthermore, transitions between different states will be reflected in spectral changes. Deviations from idealized MI and SF states22 are also measurable.

For an SF, using pSF(q) (see the Methods section), we can write the envelopes of the comb of lorentzians shown in Figs 2a,c and 3a,c. As known, the atom number at K sites fluctuates in an SF with the variance (ΔNK)2=NK(1−K/M). For example, Fig. 2c shows spectra for different lattice regions demonstrating gaussian and poissonian distributions with the spectral width , directly reflecting the atom distribution functions in an SF. For K≈M, the spectrum narrows and, for the whole lattice illuminated, shrinks to a single lorentzian as in an MI.

The condition κ<δ0=g2/Δ0a is already met in present experiments. In recent work8, where set-ups of cavity QED and ultracold gases were joined to probe quantum statistics of an atom laser with 87Rb atoms, the parameters were (g,Δ0a,κ)=2π×(10.4,30,1.4) MHz. The set-ups of cavity cooling23,24 are also very promising.

For bad cavities (κ≫δ0=g2/Δ0a), the sums can be replaced by integrals. The broad spectra in Figs 2b and 3b,d are then given by convolutions of pΨ(q) and lorentzians. For example, the blue line in Fig. 2b represents a Voigt contour, well known in spectroscopy of hot gases. Here, the ‘inhomogeneous broadening’ is a striking contribution of quantum statistics.

In summary, we showed that transmission spectra of cavities around a degenerate gas in an optical lattice are distinct for different quantum phases of equal densities. Similar information is also contained in the field amplitudes 〈a0,1〉, in contrast to previous suggestions13 that 〈a0,1〉 probes only the average density. This reflects (1) the orthogonality of Fock states corresponding to different atom distributions and (2) the different frequency shifts of light fields entangled to those states. In general, other optical phenomena and quantities depending nonlinearly on atom number operators should similarly reflect the underlying quantum statistics25,26,27.

Methods

Derivation of Heisenberg equations

A many-body hamiltonian for our system shown in Fig. 1 is given by

where a0,1 are the annihilation operators of the modes of frequencies ω0,1, wavevectors k0,1 and mode functions u0,1(r); Ψ(r) is the atom–field operator. In the effective single-atom hamiltonian Ha1, p and r are the momentum and position operators of an atom of mass ma trapped in the classical potential Vcl(r) and g is the atom–light coupling constant. We consider off-resonant scattering where the detunings between fields and atomic transition Δla=ωl−ωa are larger than the spontaneous emission rate and Rabi frequencies. Thus, in Ha1, the adiabatic elimination of the upper state, assuming linear dipoles with polarization following adiabatically, was used.

For a 1D lattice with period d and atoms trapped at xj=j d (j=1,2,…,M), the mode functions are u0,1(rj)=exp(i j k0,1xd+i φ) for travelling and u0,1(rj)=cos(j k0,1xd+φ) for standing waves where k0,1x=|k0,1|cosθ0,1, θ0,1 are angles between the mode and lattice axes and φ is some spatial phase shift (see Fig. 1).

Assuming that the modes a0,1 are much weaker than the trapping beam, we expand Ψ(r) using localized Wannier functions7 corresponding to the potential Vcl(r) and keep only the lowest vibrational state at each site (we consider a quantum degenerate gas): , where bi is the annihilation operator of an atom at site i at a position ri. Substituting this expansion in the hamiltonian H, we can get a generalized Bose–Hubbard model7 including light scattering. In contrast to ‘Bragg spectroscopy’, which involves scattering of matter waves4, and our previous work28, we neglect lattice excitations here and focus on light scattering from atoms in some prescribed quantum states.

Neglecting atomic tunnelling, the hamiltonian reads:

where . For deep lattices, the coefficients reduce to Ji,ilm=ul*(ri)um(ri) neglecting spreading of atoms, which can be characterized even by classical scattering29. The Heisenberg equations obtained from this hamiltonian are given by equation (1), where we have added a relaxation term. Strictly speaking, a Langevin noise term should also be added to equation (1). However, for typical conditions, its influence on the expectation values of normal ordered field operators is negligible (see, for example, ref. 30). Here, we are interested only in the number of photons 〈al†al〉, which is a normal ordered quantity. Thus, we can simply omit the noise term in equation (1).

Simple expressions for spectral line shapes in the SF state

We will now derive expressions for the spectra shown in Figs 2 and 3 demonstrating relations between atomic quantum statistics and the transmission spectra for the SF state. As mentioned in the main text, in all examples shown in Figs 2 and 3, the photon number depends only on a single statistical quantity, which we denote as q. Using this fact, the multinomial distribution in equation (3) reduces to a binomial, which can be directly derived from equation (3): with pSF(q)=N!/[q!(N−q)!](Q/M)q(1−Q/M)N−q and a single sum instead of M ones. Here, Q is the number of specified sites: Q is equal to K for one mode and two modes in a maximum; Q is the number of odd (or even) sites for two modes in a minimum (Q=M/2 for even M). This approach can be used for other geometries, for example, for two modes in a minimum and K<M, where equation (3) can be reduced to a trinomial distribution.

As a next approximation, we consider N,M≫1, but finite N/M, leading to the gaussian distribution with central value and width .

In high-Q cavities (κ≪δ0=g2/Δ0a), f(q) is a narrow lorentzian of width κ peaked at some q-dependent frequency, now called Δpq. As the lorentzian height is q independent, 〈f〉SF as a function of Δp is a comb of lorentzians with the amplitudes proportional to pSF(q).

Using the gaussian distribution pSF(q), we can write the envelope of such a comb. For a single mode (Fig. 2a,c and equation (4)), we find Δpq≈δ0q with the envelope

where the central frequency , spectral width and α=|η0|2/κ2. So, the spectrum envelopes in Fig. 2a,c are well described by gaussians of widths strongly depending on K.

For K→0 and K→M, the binomial distribution pSF(q) is well approximated by a poissonian distribution, which is demonstrated in Fig. 2c for K=10 and K=68. For K=M, the spectrum shrinks to a single lorentzian, as the total atom number at M sites does not fluctuate.

In other examples (Fig. 3a,c), the above expression is also valid, although with other parameters. For two modes in a diffraction maximum (Fig. 3a), the central frequency, separation between lorentzians and width are doubled: , Δpq≈2δ0q and ; α=|η0|2/(2κ2). The left satellite at Δp=0 has a classical amplitude |η0|2/(4κ2).

The non-classical spectrum for two waves in a diffraction minimum (Fig. 3c) is centred at , with components at Δpq≈2δ0q, and is very broad, ; α=|η0|2/κ2.

For bad cavities (κ≫δ0), the sums can be replaced by integrals with the same parameters and σω as for κ<δ0. For a single mode, Fig. 2b represents a Voigt contour

For two modes in a diffraction minimum, the photon number (Fig. 3d) is

where , whereas in a maximum (Fig. 3b)