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Correlations and dynamics of tunnel-coupled one-dimensional Bose gases

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Pith's one-line read Measurements of fourth-order connected correlation functions reveal non-Gaussian relative-phase fluctuations in tunnel-coupled 1D Bose gases, consistent with sine-Gordon theory, and a first observation of Gaussification after decoupling.

arxiv 1908.00422 v1 pith:HI4Q2SFV submitted 2019-08-01 cond-mat.quant-gas cond-mat.stat-mechquant-ph

Thomas Schweigler This is my paper · ORCID
classification cond-mat.quant-gascond-mat.stat-mechquant-ph
keywords phasecorrelationfluctuationsfunctionstunnelingcoherencecorrelationsdouble
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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This thesis is about what happens when two ultracold gases are placed in two parallel traps, close enough that atoms can quantum-mechanically tunnel between them. By releasing the gases and letting them overlap, the experiment records interference fringes, similar to the pattern from two laser beams. The shape of these fringes gives the phase difference between the two gases at every point along their length. By repeating the experiment many times, the researchers can compute not just the usual two-point correlations of these phase differences, but also four-point correlations.

In a system with no interactions, four-point correlations are completely determined by two-point correlations; the distribution of fluctuations is Gaussian. Here, at intermediate tunneling strength, the four-point correlations deviate from this rule, and the extra connected part, a signature of interactions between the collective sound-like excitations, becomes visible. The magnitude of this effect matches the prediction of the sine-Gordon model, a standard low-energy theory for coupled superfluids, when the model parameters are taken from independent thermometry and from the measured degree of phase locking. At zero or very strong tunneling the connected part vanishes, as the sine-Gordon model also predicts.

The thesis also studies dynamics. Starting with a non-Gaussian state in a tunnel-coupled double well, the tunneling is suddenly switched off; the phase fluctuations are then observed to evolve toward a Gaussian state, an effect called Gaussification. In separate experiments, the gas is split and re-coupled, and the relative phase oscillations are seen to damp as phase coherence builds up.

Extended reading notes

Core claim

The paper's load-bearing assertion is that, for intermediate phase locking, the fourth-order correlation function of the relative phase cannot be described by second-order functions alone and a substantial connected part remains, as stated in section 5.5.3: 'For intermediate phase locking the fourth-order function cannot be described by second-order functions alone, and a substantial connected part remains.' If correct, this establishes non-factorizing higher-order connected correlations as a signature of interactions between collective excitations in a tunnel-coupled 1D Bose gas, with a magnitude consistent with the thermal sine-Gordon model for slow-cooled preparations.

Load-bearing premise

The quantitative comparison to sine-Gordon theory assumes that the relative and common motional degrees of freedom are in thermal equilibrium at a common temperature, so that the thermal coherence length extracted from density ripple thermometry (dominated by common modes, section 4.1.3) is the correct value for the relative phase. The thesis itself questions this: 'It seems like the temperature in the relative and common degrees of freedom doesn't always coincide' (section 5.5.1). If the relative modes are hotter, the fitted parameter q and the predicted fourth-order measure would change, weakening the claimed agreement. The unresolved discrepancy in the second-order fluctuation magnitude S(2) (section 5.5.1) is a direct manifestation of this risk.

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Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central fourth-order correlation claim rests on the effective sine-Gordon description with parameters (q, lambda_T) extracted from the same experimental system, plus modeling assumptions for the imaging (Gaussian PSF) and the phase extraction. No entirely new degrees of freedom or forces are introduced. The main cost is reliance on a classical-fields effective theory whose second-order predictions show unresolved quantitative discrepancies in the same data.

free parameters (2)
  • q = lambda_T / l_J (dimensionless tunnel-coupling strength) = 0 to ~3 (range across datasets; corresponding to coherence factors from ~0.3 to ~0.96)
    Fitted from the measured coherence factor <cos(phi)> in section 5.4.3; used to generate sine-Gordon theory predictions for the fourth-order correlation measure M(4).
  • sigma_PSF (Gaussian point-spread-function width for imaging) = 3.0 um for coupled scans, 3.5 um for uncoupled scans
    Calibrated in section 5.4.2 by comparing simulated images with original phase profiles; applied to theoretical predictions before comparison with experiment.
assumptions (5)
  • domain assumption The classical fields approximation is valid for the probed modes (mode occupation beta*epsilon << 1), so thermal phase fluctuations are described by classical statistical mechanics.
    Used throughout the theoretical modeling (sections 2.2.4 and 2.2.5); justified for temperatures above ~10 nK and momenta below k_c, but breaks down near the quantum fluctuation regime.
  • domain assumption The sine-Gordon Hamiltonian (eq. 2.65) is the correct low-energy effective description for the tunnel-coupled double well in the quasicondensate regime.
    Adopted from references [66,67] and used to generate phase fluctuations (sections 2.4.2 and 5.4). Validity for the experimental parameters is discussed in section 2.5 but not derived from the microscopic Lieb-Liniger model.
  • domain assumption The relative and common degrees of freedom are in thermal equilibrium at a common temperature when extracting the thermal coherence length from density ripple thermometry.
    Stated explicitly in section 5.4.3: 'Assuming that common and relative degrees of freedom are in thermal equilibrium, we can determine the temperature T through density ripple thermometry.' The thesis later notes this may be violated (section 5.5.1).
  • ad hoc to paper The imaging process is equivalent to convolution of the phase profiles with a Gaussian of width sigma_PSF approximately 3 um.
    Used to compute theory curves (section 5.4.2); calibrated from simulated pictures rather than from first principles, and differs between coupled and uncoupled scans (3 vs 3.5 um).
  • domain assumption The phase profiles extracted from interference images after phase unwrapping faithfully represent the in-situ relative phase modulo 2 pi.
    Needed for all correlation measurements (section 4.2); checked via simulated imaging (section 5.6.2), but fails for hot systems unless the phase-slip filtering condition is applied.

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Cite this review

Pith. "Pith review of Correlations and dynamics of tunnel-coupled one-dimensional Bose gases." pith.science (2026). https://pith.science/paper/HI4Q2SFV

@misc{pith2026190800422,
  author       = {Pith},
  title        = {Pith review of: Correlations and dynamics of tunnel-coupled one-dimensional Bose gases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HI4Q2SFV}},
  note         = {Machine review of arXiv:1908.00422}
}
read the original abstract

We present a series of experiments performed with two ultracold one-dimensional Bose gases (rubidium atoms) in a double well potential. Employing matter-wave interference, we can measure the spatially resolved phase difference between the two gases and consequently investigate spatial correlations. By investigating whether higher order correlation functions can be factorized into correlations of lower order, we can investigate the interaction properties of the system. For a non-interacting system, all correlation functions with orders greater than two factorize and one observes Gaussian fluctuations. Here, we present the measurement of non-factorizing fourth-order correlation functions, leading to an experimental characterization of the interactions between the collective excitations of the quantum many-body system. The degree of non-factorizibility, i.e., the degree of non-Gaussianity of the phase fluctuations, depends on the tuneable tunneling strength between the wells. Starting from such a non-Gaussian state, we are able to observe the dynamical evolution towards a state with factorizing correlation functions (Gaussian fluctuations). We start in a double well with tunneling and then abruptly decouple the two subsystems. Subsequently, we observe how the initially non-Gaussian phase fluctuations become Gaussian. Moreover, we discuss the dynamical emergence of phase coherence in a double well potential with tunneling. We experimentally investigate the evolution starting from two different initial states. In one case, we split a cloud of atoms into two and trigger global oscillations in their relative phase. The oscillations subsequently damp and phase coherence sets in. In the other case, two independent clouds are suddenly coupled by tunneling. Again, phase coherence emerges between the two subsystems.

Figures

Figures reproduced from arXiv: 1908.00422 by the authors.

Figure 1.1
Figure 1.1. Experimental setup. (a) The two vacuum chambers of our exper￾iment are shown. In the upper chamber one can see the atomchip being mounted up-side down. Note that the picture was taken before the chambers have been sur￾rounded with coils and optics. (b) Chip-mount without atomchip. One sees the macroscopic copper structures. (c) Completed mount with the atomchip on top. Figure adapted from [36]. At the end of the MOT… view at source ↗
Figure 1.2
Figure 1.2. Wires on the atomchip. A schematic illustration showing only the wires of the atomchip which were actually used in this thesis. Figure reproduced with permission from [36]. gated trap, therefore, leads to cigar shaped clouds. In the following, we will often refer to the elongated direction (the z direction) as the ‘longitudinal’ direction. The perpendicular tightly confined directions will be the ‘trans￾verse’ direc… view at source ↗
Figure 2.1
Figure 2.1. Rescale factors for λT . (a) Shows the rescaling factor n1D h1/ρi reg (according to eq. (2.70)), for comparing the phase fluctuations of the full Hamil￾tonian (2.17) to the ones of the Luttinger liquid theory (2.16), as a function of α (2.71). (b) The red curves represent the same as is shown in (a) as a function of temperature. The solid/dash-dotted curve is for n1D = 60 and 100 µm-1 respectively. The blue lines re… view at source ↗
Figures from the paper (51 more)
Figure 3.1
Figure 3.1. Figure 3.1: Possible imaging directions. The red cigars in the figure illustrate the in situ atom clouds in a single well (a) and a double well (b) trap. After being released from the trap, the clouds expand in time of flight (TOF), as illustrated by the red blob. Subsequently a…
Figure 3
Figure 3. Figure 3: fig. 3.1). The imaging light is switched on for 75 [PITH_FULL_IMAGE:figures/full_fig_p055_3.png]
Figure 3.2
Figure 3.2. Figure 3.2: APSF with and without cloud extension for the transverse imaging. The amplitude point spread function (APSF) following from eq. (3.15) is shown. Note that the APSF is two dimensional, depending on the two coordinates y˜ and ˜z, which are perpendicular to the imaging …
Figure 3.3
Figure 3.3. Figure 3.3: Focusing the transverse imaging system: The minimum position δzmin of the g2 function (4.1) is plotted as a function of the defocus distance x0. Remember that we are imaging in the x direction here. Therefore we also denote the defocus distance with x0 instead of z0 …
Figure 3.4
Figure 3.4. Figure 3.4: APSF with and without cloud extension for the transverse imaging. Same as fig. 3.2 but with cloud extension σwf = 7.6 µm after 2 ms TOF for the red curve. 3.5.2. Resolution for small time of flight For a short TOF, the transverse width of the atom cloud is still quit…
Figure 3.5
Figure 3.5. Figure 3.5: APSF with and without cloud extension for the vertical imag￾ing. Similar to fig. 3.2 but for the vertical imaging system. The blue line shows the APSF without cloud extension, the red line with cloud extension σwf = 59.4 µm after 15.6 ms TOF. Both functions are calcu…
Figure 4.1
Figure 4.1. Figure 4.1: Influence of the imaging system onto the density ripples. (a) The g2 function (4.1) is shown without any consideration of the imaging resolution (blue line), extracted from simulated pictures (red bullets) and for the effective con￾sideration of the imaging process t…
Figure 4.2
Figure 4.2. Figure 4.2: Testing density ripple thermometry on the simulated pictures: 1000 simulated pictures have been produced for various temperatures. The g2 func￾tions (4.1) extracted from the simulated pictures were then fitted with g2 functions calculated from the Luttinger liquid mo…
Figure 4.3
Figure 4.3. Figure 4.3: Influence of the common and relative phase fluctuations onto the density ripples. (a) Illustrates how common (upper plot) and relative (lower plot) in-situ phase fluctuations lead to density ripples in TOF. The red solid and dashed lines show the in-situ phase profil…
Figure 4.4
Figure 4.4. Figure 4.4: Extraction of the relative phase. (a) The two condensates inter￾fere in 15.6 ms TOF. The picture shows the resulting interference pattern recorded through the vertical imaging system. The color encodes the atomic density, red corresponding to high density and blue to…
Figure 5.1
Figure 5.1. Figure 5.1: Atom number evolution during system preparation. The evo￾lution of the atom number N is shown as a function of the duration of evaporative cooling in a double well potential with intermediate tunnel coupling. (a) Slow cooling. Only the last 400 ms are shown, during w…
Figure 5.2
Figure 5.2. Figure 5.2: Phase locking between two clouds in a double well with fi￾nite barrier. (a) Shows the relation between fringe spacing and phase locking. The fringe spacing λF is plotted as function of the coherence factor hcos(ϕ)i. The coherence factor is calculated by averaging ove…
Figure 5
Figure 5. Figure 5: b shows the spatial dependence of the coherence factor. Note [PITH_FULL_IMAGE:figures/full_fig_p081_5.png]
Figure 5
Figure 5. Figure 5: fig. 5.3a, the independence from the temperature is only true below a certain [PITH_FULL_IMAGE:figures/full_fig_p083_5.png]
Figure 5.3
Figure 5.3. Figure 5.3: Simulated influence of the imaging system onto the cosine transformed phase fluctuations. Plotted is the simulated reduction factor for the variance of the cosine transformed phase fluctuations (see eq. (5.5)). The cosine transform was performed for the central 50 µm…
Figure 5.4
Figure 5.4. Figure 5.4: Coherence factor for the sine-Gordon model. The coherence factor hcos(ϕ)i is plotted as a function of the parameter q = λT /lJ for the thermal phase fluctuations of the classical sine-Gordon model (see section 2.4.3). The dashed gray line represents the results witho…
Figure 5.5
Figure 5.5. Figure 5.5: Testing the temperature fit with simulated pictures. The ex￾traction of the thermal coherence length λT from the relative phases according to the procedure described in section 5.4.3 is tested by applying it to pictures simu￾lated from the thermal sine-Gordon model (…
Figure 5.6
Figure 5.6. Figure 5.6: Magnitude of the relative phase fluctuations. The measure S (2) (5.6) for the magnitude of the relative phase fluctuations is plotted as a function of the coherence factor. In the upper subplot, the experimental results for the coupled slow cooled data is represented…
Figure 5.7
Figure 5.7. Figure 5.7: Cosine transformed second-order correlation function. Results for an uncoupled double well (a) and for a coupled double well (b) leading to inter￾mediate phase locking (hcos(ϕ)i = 0.80). Both measurements have been performed in a harmonic trap. The color represents t…
Figure 5.8
Figure 5.8. Figure 5.8: Variances of the cosine transformed relative phases for the uncoupled double well. The results for the cosine transformation of the central 50 µm are shown. The experimental results are corrected by the expected influence of the imaging system as discussed in the mai…
Figure 5
Figure 5. Figure 5: fig. 5.9a, the k-dependence of the variance for the cosine transformed rela [PITH_FULL_IMAGE:figures/full_fig_p092_5.png]
Figure 5.9
Figure 5.9. Figure 5.9: Variances of the cosine transformed relative phases for dif￾ferent phase locking. (a) The red bullets represent the experimental results when cosine-transforming the central 50 µm and correcting for the imaging reso￾lution (σPSF = 3 µm was used). The errorbars repres…
Figure 5
Figure 5. Figure 5: shows the experimental data for the full fourth-order correla [PITH_FULL_IMAGE:figures/full_fig_p094_5.png]
Figure 5.10
Figure 5.10. Figure 5.10: Decomposition of the fourth-order phase correlation func￾tions G(4)(z, z 0 ). Uncoupled (hcos(ϕ)i ≈ 0; a), intermediate (b) and strongly phase-locked (hcos(ϕ)i ≈ 1; (c)) regimes. To visualize the high-dimensional data, we choose z3 = −z4 = 14 µm and z 0 = 0, which r…
Figure 5.11
Figure 5.11. Figure 5.11: Relative size of the fourth-order connected correlation func￾tion. The results following from slow evaporative cooling (red bullets) and from the fast cooling procedure (blue diamond) are shown. We plot the measure M(4) (eq. (5.10)) as a function of the phase lockin…
Figure 5.12
Figure 5.12. Figure 5.12: Relative size of the fourth-order connected correlation func￾tion. The same experimental results as in fig. 5.11 are shown. In the upper subplot, the experimental results for slow cooling are marked by the red bullets. In the lower subplot, the blue diamonds mark th…
Figure 5.13
Figure 5.13. Figure 5.13: Full distribution functions and interference patterns of the phase. (a) Full distribution (probability density) functions for the phase differences ∆ϕ = ϕ(z) − ϕ(z 0 ) for z = −z 0 = 20 µm for different phase locking strengths and two different ways to prepare the q…
Figure 5.14
Figure 5.14. Figure 5.14: Phase fitting error for zero tunnel coupling. The vertical axis gives the spatial mean squared error eq. (5.11), the horizontal axis the spatial vari￾ance eq. (5.12). Each blue dot represents the result for one simulated image. For details about the simulation proce…
Figure 5.15
Figure 5.15. Figure 5.15: Phase profiles fitted from simulated pictures. The blue crosses represent the phases ϕfit(zn) fitted from the simulated images, the red crosses the corresponding input for the simulation. The quantity ϕin(zn) as discussed in the main text is plotted. Both plots repr…
Figure 5.16
Figure 5.16. Figure 5.16: Phase fitting error for intermediate phase locking. Like fig. 5.14, but for pictures simulated from the sine-Gordon theory with q = 2.9. With decreasing thermal coherence length λT , one sees the appearance of three clouds separated from the main cloud. In the main …
Figure 5.17
Figure 5.17. Figure 5.17: Phase profiles fitted from simulated pictures. Like fig. 5.15, but for sine-Gordon theory with q = 2.9 and λT = 10 µm. The uppermost plot corresponds to the point marked by the green square in fig. 5.16, the input phase profile doesn’t contain a phase slip and none …
Figure 5.18
Figure 5.18. Figure 5.18: Phase fitting error for pictures simulated from Gaussian fluc￾tuations. Same as in fig. 5.16 but for a Gaussian theory with tunnel coupling. The thermal coherence length λTrel for the relative degrees of freedom is indicated in the lower right corner of the subplots…
Figure 5.19
Figure 5.19. Figure 5.19: Phase profiles fitted from Gaussian simulated pictures. Like fig. 5.15, but for pictures simulated from Gaussian fluctuations. Red again represents the input of the simulated pictures and blue the extracted phase profile. The upper, middle and lower plot corresponds…
Figure 5.20
Figure 5.20. Figure 5.20: Influence of the imaging system onto the measure M(4) . The red bullets represent the results calculated from 1000 phase profiles fitted from simulated images. The green diamonds represent the quantities calculated from the same underlying data used to simulate the …
Figure 5.21
Figure 5.21. Figure 5.21: Measure M(4) following from pictures simulated for Gaussian fluctuations. Same as fig. 5.20, but for pictures simulated from Gaussian phase fluctuations (see section 5.6.1 for details). For very hot temperatures, the results partly coincide with the thermal sine-Gor…
Figure 5.22
Figure 5.22. Figure 5.22: Relative size of the fourth-order connected correlation func￾tion for sine-Gordon-like theories. The measure M(4) as defined in eq. (5.10) is plotted as a function of the coherence factor hcos(ϕ)i. The thermal results for sine-Gordon-like theories as defined in eq. …
Figure 5
Figure 5. Figure 5: fig. 5.23, the influence of this filtering on the measure [PITH_FULL_IMAGE:figures/full_fig_p112_5.png]
Figure 5.23
Figure 5.23. Figure 5.23: Distinguishing genuine from fake non-Gaussianity. In the upper subplot, the crosses show the measure M(4) calculated only from phase profiles fulfilling the condition (5.13) with ϕlim = π/2. For the results marked by the bullets all phase profiles are used. The expe…
Figure 6
Figure 6. Figure 6: fig. 6.1, the nominator [PITH_FULL_IMAGE:figures/full_fig_p117_6.png]
Figure 6.1
Figure 6.1. Figure 6.1: Relative size of the fourth-order connected correlation func￾tions. In the upper plot, the red bullets represent the experimental results for the measure M(4) as a function of the evolution time t. The errorbars represent 80% confidence intervals calculated by using …
Figure 6.2
Figure 6.2. Figure 6.2: Relative size of the fourth-order connected correlation func￾tion. Same as the upper subplot in fig. 6.1, but for multiple different experimental measurements. The coherence factor of the initial states is given in the upper right corner of the subplots. The two uppe…
Figure 6.3
Figure 6.3. Figure 6.3: Quench from strong phase locking. (a) The red bullets show the experimental results for the measure M(4) for the fourth-order connected part as a function of the coherence factor (see discussion in the main text). The errorbars represent the 80% confidence intervals …
Figure 7.1
Figure 7.1. Figure 7.1: Oscillations for the sine-Gordon equation without initial fluc￾tuations. Equation (7.1) is solved for ϕ(z, t = 0) = π/2 and ∂ ∂tϕ(z, t) [PITH_FULL_IMAGE:figures/full_fig_p126_7_1.png]
Figure 7.2
Figure 7.2. Figure 7.2: Circular mean phase and local oscillation frequency. (a) The circular mean ¯ϕ of the phase is shown as a function of the time t. The red and blue bullets show the experimental results for z = 0 and z = 20 µm respectively, with the errorbars giving the 80% confidence …
Figure 7
Figure 7. Figure 7: b shows the spatial dependence of the fitted [PITH_FULL_IMAGE:figures/full_fig_p130_7.png]
Figure 7.3
Figure 7.3. Figure 7.3: Mean interference pictures, for the same experimental measurement as presented in fig. 7.2. The pictures show the 2D atomic density averaged over all experimental repetitions for the evolution time stated above the plots. One sees the bending of the interference frin…
Figure 7.4
Figure 7.4. Figure 7.4: Oscillation frequency as a function of the fringe spacing. As for fig. 7.2b, we obtain ω0 from fitting the circular mean phase with the solution of the damped pendulum. However, here we do not fit the local ¯ϕ but the circular mean phase also averaged over the centra…
Figure 7.5
Figure 7.5. Figure 7.5: Evolution of the integrated contrast for different start phases. The squared integrated fringe contrast c 2 (L) calculated according to eq. (7.6) is shown as a function of time. The integration length is L = 42 µm. The bullets represent the experimental results for d…
Figure 7.6
Figure 7.6. Figure 7.6: Universal evolution of the coherence factor. The bullets represent the experimental data for measurements with the same starting phase but different double well separations leading to different tunnel coupling strengths. The solid lines are a guide to the eye. On the…
Figure 7.7
Figure 7.7. Figure 7.7: Evolution of the coherence factor after recoupling. Tunneling between two independent condensates is switched on by ramping down the double well barrier in 2 ms. The time t = 0 corresponds to the end of the ramp. The bullets represent the experimental results, with t…
Figure 7.8
Figure 7.8. Figure 7.8: Mean interference pictures. Similar to the mean interference pic￾tures in fig. 7.3, but for the results corresponding to the blue curve in fig. 7.7. One sees that at time t = −2 ms, before the double well barrier is ramped down, the relative phase is completely rando…

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