Proposal for interferometric detection of the topological character of modulated superfluidity in ultracold Fermi gases

As a fascinating example of self-organized quantum matter, Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states have long been sought after in exotic superconductors [1, 2] and in ultracold atomic gases [3], and they may even occur in neutron stars [4]. In superconductors, modulated superfluidity results from the interplay between superconducting pairs and an applied parallel magnetic field. The inter-electron attraction favors a superfluid (SF) state consisting of pairs of up- and down-spin electrons, whereas the field favors a polarized Fermi liquid (FL) state with a lower Zeeman energy. The simplest theories predict a first-order SF–FL transition [5, 6]. However, the competition between pairing and polarization can produce far more subtle physics—an intermediate FFLO state [7–12], henceforth referred to as LO⁴, as depicted in figure 1. As the field is increased beyond h_c1 it forces excess fermions into the SF in the form of domain walls where the order parameter changes sign between positive and negative values. (This is analogous to how a perpendicular field forces vortices into a superconducting film.) The wavelength of these modulations decreases with increasing field and ultimately the system gives way to a polarized Fermi liquid.

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Cold atom experiments provide a highly tunable testing ground for finding LO states, and there have been several proposals to do so through modulations of the polarization in real space [12], peaks in the pair momentum distribution at the modulation wavevector [13, 14], shadow features in the single-particle momentum distribution [12], and Andreev bound states in the density of states [12]. A recent experiment on an array of one-dimensional (1D) tubes found density profiles in agreement with Bethe ansatz calculations [15], which predict power-law LO correlations at zero temperature. However, direct evidence of the sign changes of the order parameter—the defining feature of an LO state—is still lacking.

In this paper, we propose an interference experiment in which two isolated SFs expand into each other, as illustrated in figure 2. The ideal situation for detecting the LO phase is shown in the lower panel of this figure, where one layer is a uniform fully-paired SF, which serves as a reference phase, and the other layer is a modulated (LO) SF. The resulting interference pattern is directly sensitive to real-space modulations of the order parameter, and should provide an unequivocal signature of the elusive LO phase.

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LO states have been predicted to exist in various situations [12, 16, 17]. The most likely of these to be realized in the near future is an array of tubes with a small intertube coupling [13, 18, 19]. This quasi-1D geometry provides good Fermi surface nesting at the LO wavevector q_LO = k_F↑ − k_F↓, together with Josephson coupling between the order parameter in adjacent tubes which is necessary to stabilize true long-range order.

Therefore, it seems that the idea can be best realized in the following way. Two species of fermions (referred to as ↑ and ↓) are loaded into an optical trap. The cloud is separated into two independent quasi-2D 'pancake' layers using an optical lattice with a wide spacing in the vertical direction z (created with two laser beams intersecting at a shallow angle [20]). In order to generate an LO phase it is necessary for the two layers to have different relative population imbalances. This may happen by chance due to natural number fluctuations, or alternatively, one can induce hyperfine transitions using Raman transitions, where the two lasers are detuned at the appropriate RF frequency. In order to address each layer separately, it would be necessary to make the beam diameters smaller than the interlayer distance. In this paper, we do the analysis of the interference patterns assuming that one of the layers is balanced while the other contains a population imbalance, but this is not necessary. As long as there are different relative population imbalances between the two layers, LO signatures will be observable. An additional optical lattice is further used to create a 2D array of weakly-coupled tubes, conducive to the formation of an LO state. The resulting geometry is shown in figure 3.

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Once the gases are allowed to equilibrate for sufficient time, the interactions in the system are quickly ramped to the Bose–Einstein condensate (BEC) side of the Feshbach resonance [21] to 'freeze' or 'project' the pair wavefunction into a boson wavefunction, so that from this point onwards the pairs move as independent bosons (instead of disintegrating into fermions). Then, the confining potentials are abruptly turned off. As the clouds expand into one another, they interfere and form a 3D matter wave interference pattern. The projection of this interference pattern onto the x-z plane can be measured by absorption of a resonant probe beam along the y direction. Any LO phase modulation features will be captured in these projected interference patterns. It may be desirable to image the minority (↓) atoms, which gives the final distribution of the tightly bound pairs without contamination from unpaired majority (↑) atoms.

An earlier paper [22] has discussed the interferometric detection of the FFLO phase through correlation functions [23, 24] obtained by averaging over many interference snapshots. We emphasize that our proposal considers the information in the individual snapshots themselves, most notably the 'tire-tread' staggered interference fringes corresponding to the sign change of the paring amplitude across the domain walls. The information in these individual snapshots contains distinct information from that in the correlation functions. For example, if the domain wall spacing is not uniform in multiple snapshots, possibly due to density fluctuations between different tubes, the LO information could be lost in the averaging necessary to obtain correlation function data.

We now discuss analytically the salient features of the interference patterns of a 2D array of coupled tubes. We begin by considering two layers, each containing N coupled tubes, with separation d in the z-direction. After ramping up the interaction to produce a molecular BEC (of fermion pairs), the wavefunction is

$$\begin{eqnarray} &&\fl \Delta(x,y,z) = \mathrm{e}^{- (z - d/2)^2/2 \sigma_z^2} \sum_{n=1}^N \Delta^{\mathrm{T}}_n(x)\, \mathrm{e}^{-(y - an)^2/2 \sigma_y^2} + \mathrm{e}^{- (z + d/2)^2/2 \sigma_z^2} \sum_{n=1}^N \Delta^{\mathrm{B}}_n(x)\, \mathrm{e}^{-(y - an)^2/2 \sigma_y^2},\nonumber\\ \end{eqnarray}\noindent \tag{ 1 }$$

where a is the separation between in-plane tubes and σ_y and σ_z are the Gaussian confinements in the respective directions. The wavefunction in the nth tube is denoted by Δ^T_n(x) in the top layer and by Δ^B_n(x) in the bottom layer. When the trap and lattices are switched off, the clouds expand predominantly in the tightly confined directions (y and z). After a suitable time-of-flight t, the wavefunction is effectively Fourier-transformed in the y and z directions. That is, the final wavefunction Δ(x,y,z,t) is approximately proportional to the initial momentum distribution in the y and z directions, i.e. Δ(x,y,z,t) ≈ Δ(x,k_y,k_z,t = 0) where y = tk_y/m and z = tk_z/m:

$$\begin{eqnarray} &&\fl \Delta(x,k_y,k_z) = \sigma_y \sigma_z\, \mathrm{e}^{- z^2 \sigma_z^2/2}\mathrm{e}^{-y^2 \sigma_y^2/2} \left[\mathrm{e}^{\mathrm{i} k_z d/2} \sum_{n=1}^N \Delta^T_n(x)\, \mathrm{e}^{\mathrm{i}k_yan} + \mathrm{e}^{-\mathrm{i} k_z d/2} \sum_{n=1}^N \Delta^B_n(x)\, \mathrm{e}^{\mathrm{i}k_yan} \right].\nonumber\\ \end{eqnarray} \tag{ 2 }$$

The 3D density of the cloud, after expansion, is given by I(x,k_y,k_z)∝|Δ(x,k_y,k_z)|². The imaging process measures the integrated density along the y direction, $I(x, k_z) \propto \int \mathrm {d}k_y | \Delta (x, k_y, k_z) |^2$:

$$\begin{eqnarray} &&\fl I(x, k_z) \propto\mathrm{e}^{-\sigma_z^2 z^2} \sum_{n=1}^N\sum_{m = 1}^N\mathrm{e}^{-(n-m)^2 a^2/4 \sigma_y^2} \left[ \Delta^T_n(x) \Delta^T_m(x) + \Delta^B_n(x)\Delta^B_m(x) + 2 \Delta^T_n(x) \Delta^B_m(x) \cos k_z d \right].\nonumber\\ \end{eqnarray}\noindent \tag{ 3 }$$

We can consider the behavior of the above interference formula in its two limits: widely separated tubes (a/σ_y → ∞) and overlapping tubes (a/σ_y → 0). In the overlapping limit, the total interference pattern reduces to that of two isolated 2D layers. In the limit of widely separated tubes, similar to the proposed experiment, the total interference pattern reduces to the sum of the interference patterns from adjacent tubes in the two layers.

Realistic parameters are interlayer spacing d = 3 μm, layer thickness σ_z ≈ 200 nm, and optical lattice spacing 532 nm in the y direction [15, 20]. This lattice should be shallow enough to allow sufficient intertube hopping to lock the position of domain walls between adjacent tubes. Our analysis is valid when the domain wall spacing is much larger than σ_z, which is true for small imbalances.

The above analysis involves just two layers, which are necessary and sufficient for generating interference. Introducing more layers would complicate the analysis, and reduce the visibility of the interference pattern⁵; nevertheless, there would still be observable effects of the order of $1/\sqrt {N_{\mathrm {layers}}}$.

Figure 4 illustrates the projected interference pattern, described by equation (3), for three configurations of the upper layer: a uniform SF, an LO phase with domain walls locked between tubes, and an LO phase with domain walls fluctuating between tubes. In each case we assume that the lower layer has been prepared in a uniform SF state. The form of the LO state that we use is the Jacobi elliptic function sn(x|k), which is a good approximation of the pair wavefunction in the limit of quasi-1D tubes with small intertube coupling [25]. Each wavefunction is multiplied by Gaussian envelopes in the x, y, and z directions to mimic the effect of the trapping potential. The staggered fringe pattern in the lower two panels is a clear signature of oscillations of the relative phase between the SF and LO layers, in contrast to the straight interference fringes in the top panel.

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To better understand how the interference pattern evolves once the trapping potentials have been switched off, we have simulated the time-of-flight evolution of our experimental setup. To do this, we numerically evolve the free-particle Schrödinger equation

$$\begin{equation} \frac{\partial}{\partial t} \Delta(x,y,z,t) = \frac{\mathrm{i} \hbar}{2\,m} \nabla^2 \Delta(x,y,z,t) \end{equation} \tag{ 4 }$$

subject to the initial pairing amplitude when the trap is abruptly turned off (t = 0).

We have analyzed the evolution of the interference pattern for a two-tube geometry similar to what is shown in the right panel of figure 2, where one of the tubes is in a fully paired SF state and the adjacent tube is in an LO state. The LO state we used is the same as that used in the analysis of figure 4 above. This provides the opportunity to test the validity of the approximation that we need to only consider expansion in the two transverse directions, and that a suitable time-of-flight can be found in experiments that achieves this 'far field' limit. The results of our simulation are shown in figure 5. To set the time and length scales, we have set the length of our tubes to be approximately 50 μm, and used the mass of ⁶Li, typical values for such an experiment [15]. The spacing of the domain walls is about 10 μm, which is also the spacing of the bright and dark interference signals in the horizontal direction. This length scale is accessible in similar time-of-flight experiments [26] that have a resolution of ≈3 μm. In the central panel of figure 5, we see that there is a suitable intermediate time-of-flight where expansion in the longitudinal (x) direction can be neglected, and the interference patterns shown in figure 4 are valid. The expansion time of this snapshot is t ≈ mdR_⊥/ℏ, where d is the separation of the tubes and R_⊥ is the transverse radius of the tube, can be associated with the 'far field' limit of expansion in the transverse (z) direction [27].

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Our simulations also allow us to comment on the effects of expansion along the x-axis in our experimental proposal. In the final frame of figure 5, we show the interference pattern after long time expansion. Despite the fact that the analysis of section 3 for the interference pattern breaks down in this regime, the LO interference signatures remain visible.

Our time-of-flight simulation contains two isolated tubes for computational simplicity. In the experiment proposed above, we have considered an array of coupled tubes in order to quench the fluctuations. The time-of-flight calculation can be extended to an array of tubes, similar to the proposed experiment. In this geometry, the expansion in the y-direction would approach the far-field limit at times t_y ≈ maR_⊥/ℏ. Since a ≈ d/6, t_y < t_z = mdR_⊥/ℏ which implies that when the far-field limit is satisfied in the z-direction, the far-field limit is also satisfied in the y-direction, and the analysis discussed in section 3 is valid.

Even with intertube coupling to stabilize quasi-long-range-order, we expect thermal and quantum fluctuations to have an impact on the interference patterns in our proposed experiment.

5.1. Thermal fluctuations

The interference experiment that we have proposed essentially takes 'snapshots' of the wavefunction after the time-of-flight. For this reason, the thermal phase fluctuations are not integrated over, and can be considered as domain wall fluctuations between tubes. However, as shown in the lowest panel of figure 4, these fluctuations are not severe enough to destroy the occurrence of staggered interference fringes, the signature of an LO phase. This is further detailed in the bottom right panel of the same figure, where we show a cross section of the interference pattern.

A remarkable feature of cold atom experiments is that control parameters (interaction, lattice depth, and trap depth) can be turned off very quickly, much faster than the typical timescale of domain wall movement. This allows us to take a snapshot of the wavefunction (resolved in real time), in a way not possible in condensed matter experiments. Thus, even above the critical temperature where thermal fluctuations destroy long-range order, it may still be possible to detect LO physics in the form of 'temporary' domain walls!

In order to study the effect of thermal fluctuations quantitatively, we have performed Bogoliubov–de Gennes (BdG) simulations of the 2D attractive Hubbard Hamiltonian

$$\begin{equation} H = -\sum_{\left<\mathbf{r},\mathbf{r'} \right>, \sigma}{t_{\mathbf{r} \mathbf{r'}} (c^\dagger_{\mathbf{r} \sigma} c_{\mathbf{r'} \sigma} + c^\dagger_{\mathbf{r'} \sigma} c_{\mathbf{r} \sigma})} - \sum_{\mathbf{r}, \sigma} {(\mu_\sigma - V_{\mathbf{r}}) n_{\mathbf{r} \sigma}} \end{equation} \tag{ 5 }$$

$$\begin{equation} \hspace{2pc} - \left|U\right| \sum_{\mathbf{r}} {\left(n_{\mathbf{r} \uparrow} - \frac{1}{2}\right)\left(n_{\mathbf{r} \downarrow} - \frac{1}{2}\right)} \end{equation} \tag{ 6 }$$

with hopping t_rr' = t for nearest-neighbor bonds along the length of the tube (in the x direction), t_rr' = t_⊥ for nearest-neighbor bonds between tubes, fermion creation and annihilation operators c^†_rσ and c_rσ, number operators n_rσ = c^†_rσc_rσ, chemical potentials μ_σ = μ − σh for the two spin species, Zeeman field h, local Hubbard attraction $\left |U\right |$, and confining harmonic potential V_r.

The results of these simulations on a system of 7 tubes with 50 sites in each tube are shown in figure 6. The harmonic potential we consider mimics the experimental proposal. The behavior of the BdG pairing amplitude $\Delta (\mathbf {r}) \propto \left < c_{\mathbf {r} \uparrow } c_{\mathbf {r} \downarrow } \right >$ in the central three tubes as a function of temperature for intertube coupling strengths t_⊥ = 0.1t and 0.3t is shown. For small intertube coupling t_⊥/t = 0.1, we see domain wall fluctuations between the tubes whose effect we have included in our interference pattern analysis in figure 4. As the intertube coupling is increased at low temperatures, the phases of the domain walls in the different tubes 'lock' together. Our BdG results show that the LO state survives up to a temperature of T_LO ≈ 0.12t. We expect phase fluctuations to suppress T_LO and to also depend on the intertube coupling; the larger the coupling, the smaller the suppression. However, as we have emphasized above, in spite of these fluctuations, the interferometric signature would still be visible up to T* ∼ T_LO (which is a significant fraction of the temperature for the onset of pairing T_BCS ≈ 0.2t for these parameters).

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5.2. Quantum fluctuations

At low temperatures, the excess fermions residing in the domain walls quantum tunnel from one position to another. This can be viewed as the diffusion of domain walls in imaginary time τ; measurements are necessarily averaged over τ. For isolated tubes, quantum fluctuations of the domain walls prevent long-range LO order, and there is only quasi-long-range order at zero temperature; hence, the interference pattern will be washed out. This is why we recommend using sufficient coupling between the tubes to lock their phases so that the pairing amplitude modulations remain even after averaging over quantum fluctuations (see figure 7).

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5.3. Stabilization of fluctuations with intertube coupling

5.4. Conclusions

We gratefully acknowledge useful discussions with Randy Hulet. This work was supported by ARO grant numbers W911NF-08-1-0338 (NT) and DARPA grant no. 60025344 under the Optical Lattice Emulator (OLE) program (YLL and NT). MS acknowledges support from the NSF Graduate Research Fellowship Program.