Abstract
The dynamic structure factor is a central quantity describing the physics of quantum manybody systems, capturing structure and collective excitations of a material. In condensed matter, it can be measured via inelastic neutron scattering, which is an energyresolving probe for the density fluctuations. In ultracold atoms, a similar approach could so far not be applied because of the diluteness of the system. Here we report on a direct, realtime and nondestructive measurement of the dynamic structure factor of a quantum gas exhibiting cavitymediated longrange interactions. The technique relies on inelastic scattering of photons, stimulated by the enhanced vacuum field inside a high finesse optical cavity. We extract the density fluctuations, their energy and lifetime while the system undergoes a structural phase transition. We observe an occupation of the relevant quasiparticle mode on the level of a few excitations, and provide a theoretical description of this dissipative quantum manybody system.
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Introduction
An interacting quantum manybody system can be characterized by analysing its response to a weak perturbation. In the framework of linear response theory, a key quantity is the dynamic structure factor, which is the Fourier transform of the spatial and temporal density–density correlations^{1,2}. Knowledge of the dynamic structure factor provides a complete picture of the emerging quasiparticle modes^{3}, their excitation energy, lifetime and mean occupation number. These quasiparticle modes determine the collective density fluctuations of the system and may also characterize the critical behaviour in the vicinity of a phase transition^{4}. For example, in a longrange interacting system, a structural phase transition can be driven by a rotonlike mode softening^{5,6,7,8}, which is expected to show up as a thermally enhanced peak in the dynamic structure factor^{9}.
In solid state systems, the dynamic structure factor S(k,ω) can be measured by illuminating a sample with a beam of neutrons^{10} or Xrays^{11}, and analysing the inelastically scattered particles with regard to their change in energy ℏω and momentum ℏ k. In dilute quantum gases, a measurable signal has only been obtained from photons elastically scattered off a densitymodulated sample^{12,13,14,15}. Direct detection of inelastically scattered photons into free space^{16}, in analogy to neutron scattering, is however hindered by a vanishingly small signal^{17}, see Fig. 1a.
A technique measuring the spectral response function of a quantum gas, that is, the dynamic structure factor at zero temperature^{18}, is Bragg spectroscopy^{19,20}. It is based on stimulated rather than spontaneous inelastic scattering of photons between two laser beams. Therefore, the transfer of momentum and energy to the atomic cloud is predetermined by the angle and frequency difference between the beams, and is typically measured via destructive absorption imaging. A complementary detection method analyses the change in light field intensity in one of the two Bragg beams^{21} and could in principle be extended with the help of cavities to be only weakly perturbative^{22}. However, all these methods measure the linear response of the gas on a perturbation and are insensitive to thermally excited quasiparticles. A different approach, in situ imaging, has been used to extract the temperaturedependent static structure factor S(k)=∫dωS(k,ω) of a twodimensional gas^{23}. Similar to the analysis of noise correlations from images of ballistically expanded ultracold gases^{24}, this approach gives no access to the quasiparticle spectrum, that is, the temporal dynamics.
Here we present a nondestructive, direct measurement of the dynamic structure factor at distinct kvectors in a Bose–Einstein condensate (BEC) undergoing a structural phase transition induced by cavitymediated longrange interactions. We place a BEC into an ultrahigh finesse optical cavity^{25,26} and illuminate the atoms with a transverse laser field. The enhanced vacuum field inside the optical resonator^{27} increases the spontaneous inelastic scattering of photons into the cavity mode by several orders of magnitude, so that photons leaking out of the cavity mode give rise to a detectable signal and access to the density correlations in real time, see Fig. 1b. Therefore, density fluctuations in the gas are mapped onto fluctuations of the light field, which then can be directly accessed.
Results
System description
In our experiment, the transverse laser field acts simultaneously as a pump field controlling the longrange interactions^{8,28}. This field has a frequency ω_{p} and a wavevector k_{p}, and is in a standingwave configuration directed perpendicularly to the cavity mode. It is far detuned from atomic resonance to avoid electronic excitation of the atoms. At the same time, it is detuned by only a few cavity linewidths from the cavity resonance, which enables vacuumstimulated scattering of pump photons into the cavity mode at wavevector k_{c}. These twophoton processes mediate longrange atom–atom interactions in the BEC, giving rise to a rotonlike mode softening^{8} and a structural phase transition^{28,29}. The same twophoton processes are exploited for detection. The light scattered from the transverse pump field into the cavity mode can be regarded as a superposition of all field amplitudes scattered by the individual atoms. It thus carries information on the density–density correlations of the gas at the wavevectors k_{cb}=±k_{p}±k_{c}, which are determined by the underlying twophoton processes^{29}. Within the cavity linewidth, which is two orders of magnitude larger than the frequency of the relevant quasiparticle excitation, the energies of the photons stimulated into the vacuum mode are not fixed. This is in contrast to Bragg spectroscopy, where the energy of the quasiparticle excitations that are created during probing is determined by the frequency difference of the classical fields driving the twophoton processes. The spectral analysis of the light field leaking out of the cavity thus gives us direct access to the dynamic structure factor at finite temperatures.
As previously described^{8,28,29,30}, we trap a BEC of N=1.0(1) × 10^{5 87}Rb atoms at the centre of an ultrahighfinesse optical Fabry–Pérot cavity and illuminate it by the transverse pump field. The cavitymediated interaction leads to the formation of a quasiparticle mode, which is a superposition of the collective momentum excitation at wavevectors k_{cb} of the BEC and a tiny admixture of photons inside the cavity (see Supplementary Note 1). Neglecting atom–atom collisions, its energy ℏω_{s} equals in the limit of zero pump power P the bare energy of a single momentum excitation and decreases with increasing power^{8}, where m denotes the atomic mass. Owing to this mode softening, the energy ℏω_{s} of the quasiparticle mode approaches zero at a critical pump power P_{cr}, which leads to a phase transition from a normal state with a flat density distribution to a selforganized state with checkerboard density modulation. The emergent density structure leads to elastic scattering of transverse pump light and a macroscopic population of the cavity mode. An order parameter of this phase transition is the expectation value of the operator describing the overlap of the atomic density and the checkerboard mode structure, .
Dynamic structure factor
As the cavity decay rate κ=2π × 1.25 MHz is more than two orders of magnitude faster than the evolution rate ω_{s} of the coupled system, the light field inside the cavity adiabatically follows the order parameter, (ref. 29). The frequency spectrum of the light leaking out of the cavity thus reveals the temporal and spatial Fourier transform of the atomic density correlations, evaluated at one particular wavevector. Specifically, the dynamic structure factor of the system at wavevector k_{cb} is related to the cavity field according to
where ω is the frequency shift of the cavity output field from the pump light frequency ω_{p} because of inelastic scattering. PSD(ω) is the power spectral density of the intracavity light field with the mean coherent field amplitude , and η is the twophoton Rabifrequency of the scattering process, proportional to . is the detuning between the pump laser frequency and the dispersively shifted cavity resonance (see Supplementary Note 1).
To analyse the light field leaking out of the cavity, we use a balanced heterodyne detection scheme^{30}. Figure 2 shows the power spectral density PSD(ω) of the light field as we linearly increase the transverse pump power P across the critical point. The rate of change of P/P_{cr} is a few Hertz, such that the system can be assumed to adiabatically follow its steady state throughout the measurement^{29,30}. The small panels in Fig. 2 show examples of S(k_{cb},ω) for different values of P/P_{cr}, converted via equation (1) and normalized to unity for the noninteracting case^{31}. The data reflect the microscopic processes taking place: pump photons of frequency ω_{p} inelastically scattered at the atomic ensemble will be shifted in their frequency. They become visible as red (blue) sideband at frequency ω_{p}−ω_{s} (ω_{p}+ω_{s}) if they create (annihilate) a quasiparticle. We observe the corresponding sidebands whose frequency shift tends to zero when approaching the critical point at P/P_{cr}=1 from either side of the phase transition. These density fluctuations can be distinguished from a checkerboard density modulation at k_{cb} at which pump photons will be elastically scattered without a frequency shift, visible at ω=0. Intuitively, this light field arises from scattering at Bragg planes in the densitymodulated cloud. The transverse pump power P influences not only the effective longrange interactions in the system but also the measurement process itself. For increasing pump power, the measurement imprecision due to the shot noise of the transverse pump field becomes less relevant, as can be seen from the decreasing background level of S(k_{cb},ω)^{32}. The influence of the inevitable measurement backaction will be discussed further below.
The total power of elastically scattered light is proportional to the square of the density modulation and is displayed as open symbols in Fig. 3. This coherent density modulation increases over five orders of magnitude while crossing the critical point. In the normal phase (P/P_{cr}<1), we also observe a weak field at the pump laser frequency (that is, at ω=0), which originates from a small symmetrybreaking field caused by the finite size of the system and residual scattering of the transverse pump beam at the cavity mirrors^{29,30}.
The total power of frequencyshifted light is proportional to the variance of the checkerboard density fluctuation , and thus to the static structure factor S(k_{cb})=∫dω S(k_{cb},ω). We show the variance of density fluctuations as filled symbols in Fig. 3 and observe a divergence when approaching the critical point P/P_{cr}=1 from either side, heralding a secondorder phase transition. The inset displays the variance of the density fluctuations on a double logarithmic plot to illustrate the scaling behaviour. The variance is plotted as a function of the Hamiltonian coupling parameter λ, derived from measured quantities and using a theoretical model (see Supplementary Note 1). In the normal and the selforganized phase, we extract critical exponents of 0.7(1) and 1.1(1), respectively. We attribute the deviation from our previous measurement in the normal phase, which gave 0.9(1) (ref. 29), to the refined model used for the scaling of the horizontal axis and the improved measurement scheme that allows us to directly distinguish between density fluctuations and a density modulation. Current theoretical research taking into account the open character of the system due to cavity dissipation predicts an exponent of 1.0 (refs 33, 34). Also the influence of thermal noise was theoretically shown to lead to an exponent of 1.0 (refs 35, 36). The difference between the experimentally observed exponent and the predicted value might originate from finite size effects and the presence of a small symmetrybreaking field. Further, the theory models do not include damping of the momentum excitation because of atom–atom collisions^{37,38,39}.
Characterization of the quasiparticle mode
The access to the dynamic structure factor S(k_{cb},ω) allows us to characterize the quasiparticle mode that emerges because of the longrange interactions in the gas. When adiabatically switching on the longrange interactions (P≠0), new quasiparticle modes of polaritonic character form, where intracavity photons are admixed to the recoil momentum states. A diagonalization of the Hamiltonian for P≠0 leads to the definition of a quasiparticle mode for the interacting system with annihilation and creation operators and , respectively (Supplementary Note 1). From our measurements (Fig. 2), we can directly extract the energy ℏω_{s} of this quasiparticle mode as a function of P/P_{cr}. To this end, we fit a resonance curve of a damped harmonic oscillator to both sidebands of S(k_{cb},ω) (Supplementary Note 2), whose peak positions correspond to ω_{s}, see Fig. 4. We observe the mode softening towards the critical point from both sides of the phase transition. The width γ of the sidebands is displayed in the lower panel of Fig. 4 and characterizes the damping of the quasiparticle mode. For our parameters, the main constituent of the quasiparticle mode is the atomic component, while the light field is only weakly admixed. We thus attribute the observed damping mainly to the decay of atomic momentum excitations. The finite decay rate κ of the cavity light field gives rise to an additional damping of the quasiparticle mode estimated to be only a few Hertz (Supplementary Note 1). The behaviour of the damping rate γ has been studied theoretically and originates from a resonant enhancement of the Beliaev damping of the checkerboard density wave^{37,38,39} and from finite temperature effects^{36}. Our characterization of the quasiparticle mode is consistent with earlier measurements^{8,29}, but now also extends into the organized phase because we can distinguish density fluctuations from a density modulation.
Occupation of the quasiparticle mode
The occupation of the quasiparticle mode can be extracted from the observed sideband asymmetry. As can be seen in Fig. 2, the redshifted sideband dominates over the blueshifted one. For a system in its ground state, only creation processes are possible, leading to a vanishing blueshifted sideband. This has been used for thermometry of trapped ions and cavity optomechanical systems^{40,41}. From the observation of the finite blueshifted sideband in our experiment, we infer that the system is in a steady state close to its ground state. The continuous measurement process via cavity decay constantly creates and annihilates quasiparticles at rates and . Here is the integrated spectral weight of the blue (+) and red (−) sidebands, respectively. This measurement backaction effectively gives rise to a heating rate of the system, and thus to a finite occupation of the quasiparticle mode. At the same time, the finite decay rate γ of the quasiparticle mode changes this occupation. On one hand, quasiparticles will be annihilated because of this dissipation channel at rate . On the other hand, it couples the quasiparticle mode to a thermal heat bath provided by the atomic cloud. This creates quasiparticles at rate , where is the thermal occupation of the quasiparticle mode calculated from the Bose distribution function, and T=38(10) nK is the temperature of the BEC, measured independently from absorption images. In steady state, the different contributions are balanced according to the rate equation (see Supplementary Note 1 and Supplementary Fig. 1),
We can determine the occupation of the quasiparticle mode, using the weights of the sidebands and the dissipation rates γ (empirical fit) and κ (see Fig. 5). We observe an average occupation of the quasiparticle mode on the level of only a few quanta. In the organized phase an increase in the mode occupation towards the critical point seems visible. The occupation is expected to diverge when approaching the critical point since the energy of the soft mode vanishes and the atomic damping rate goes to zero^{38,39}. This situation is very similar to the enhanced thermal occupation of rotonlike states predicted for dilute quantum gases with dipolar interactions at finite temperatures^{9}.
Discussion
We used vacuumstimulated scattering of light to directly measure the dynamic structure factor of a quantum gas with cavitymediated longrange interactions. Access to the dynamic structure factor allowed us to characterize the relevant quasiparticle mode while the system crossed a structural phase transition, to distinguish density modulation and density fluctuations, and to measure the critical exponents of the density fluctuations. We further extracted the finite occupancy of the quasiparticle mode under the influence of measurement backaction because of cavity decay and an atomic bath at finite temperature. While this measurement was motivated by the very specific setup used to create cavitymediated longrange interactions, an extension to more general settings seems possible. The approach of applying quantum optical methods based on strong matter–light interaction to the investigation of dilute ultracold gases offers unique possibilities for the nondestructive realtime investigation of quantum matter and its phase transitions^{22,34,42,43,44}.
Methods
Experimental procedure
After centring an almost pure ^{87}Rb BEC trapped in a crossedbeam dipole trap with respect to the TEM_{00} cavity mode, the transverse pump power P, at wavelength λ_{p}=785.3 nm, is increased over 100 ms to a relative coupling strength of P/P_{cr}≈0.46. Subsequently, the power P is linearly increased over 0.5 s to P/P_{cr}≈1.38, while the stream of photons leaking out of the cavity is detected in a balanced heterodyne configuration using a local oscillator power of 2.2 mW and balanced photodiodes (Thorlabs PDB110A)^{30}. The extracted quadratures at a beat frequency of 59.55 MHz are mixed down to 50 kHz, amplified, lowpassfiltered and digitized using highspeed analoguetodigital converters with 2 μs resolution (National Instruments PCI6132). The response of the heterodyne system is 2.2 V^{2} per cavity photon.
The temperature of the initially prepared BEC was determined from absorption images to be T=20(10) nK. The faroff resonant transverse pump beam heats the BEC during probing to a temperature of T=38(10) nK at the critical point. Residual atom loss of 26% during probing is included by rescaling the relative coupling axis according to the proportionality P_{cr}∝N^{−1}.
The phase transition point is characterized by a steep increase in the intracavity photon field. We fit the rise of the photon field once it has first exceeded a mean intracavity photon number of 4.5 with a saturation function p_{0} × (1−t_{cr}/t)^{p1}. With the extracted occurence time t_{cr} of the phase transition, we can convert the time axis into a relative coupling axis P/P_{cr}. The relative statistical error of P_{cr} according to this procedure is given by 5 × 10^{−4}, including intensity fluctuations of the transverse pump.
Additional information
How to cite this article: Landig, R. et al. Measuring the dynamic structure factor of a quantum gas undergoing a structural phase transition. Nat. Commun. 6:7046 doi: 10.1038/ncomms8046 (2015).
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Acknowledgements
We acknowledge insightful discussions with C. Chin, R. Chitra, S. Diehl, N. Dogra, L. Hruby and S. Huber. This study was supported by Synthetic Quantum ManyBody Systems (European Research Council advanced grant), the EU Collaborative Project TherMiQ (Grant Agreement 618074) and the DACH project ‘Quantum Crystals of Matter and Light’.
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R.L., R.M. and F.B. performed the experiments, R.L., R.M., F.B. and T.D. analysed the data. All authors contributed to the design of the experiments and the writing of the manuscript.
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Supplementary Figure 1, Supplementary Notes 13 and Supplementary References (PDF 640 kb)
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Landig, R., Brennecke, F., Mottl, R. et al. Measuring the dynamic structure factor of a quantum gas undergoing a structural phase transition. Nat Commun 6, 7046 (2015). https://doi.org/10.1038/ncomms8046
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DOI: https://doi.org/10.1038/ncomms8046
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