Simulating a quantum commensurate-incommensurate phase transition using two Raman-coupled one-dimensional condensates

We study a transition between a homogeneous and an inhomogeneous phase in a system of one-dimensional, Raman tunnel-coupled Bose gases. The homogeneous phase shows a flat density and phase profile, whereas the inhomogeneous ground state is characterized by periodic density ripples and a soliton staircase in the phase difference. We show that under experimentally viable conditions the transition can be tuned by the wave-vector difference $Q$ of the Raman beams and can be described by the Pokrovsky-Talapov model for the relative phase between the two condensates. Local imaging available in atom chip experiments allows us to observe the soliton lattice directly, while modulation spectroscopy can be used to explore collective modes, such as the phonon mode arising from breaking of translation symmetry by the soliton lattice. In addition, we investigate regimes where the cold atom experiment deviates from the Pokrovsky-Talapov field theory. We predict unusual mesoscopic effects arising from the finite size of the system, such as quantized injection of solitons upon increasing $Q$, or the system size. For moderate values of $Q$ above criticality, we find that the density modulations in the two gases interplay with the relative phase profile and introduce novel features in the spatial structure of the mode wave functions. Using an inhomogeneous Bogoliubov theory, we show that spatial quantum fluctuations are intertwined with the emerging soliton staircase. Finally, we comment on the prospects of the ultracold atom setup.

Figure 1

Experimental platform: Raman tunnel coupled Bose gases two Raman laser beams R1 and R2 with wave vectors ${\vec{k}}_1$ and ${\vec{k}}_2$ induce a spatially modulated tunnel coupling between two Bose gases ${\psi }_1\left(x\right)=\sqrt{n_1\left(x\right)}{e}^{i{\theta }_1\left(x\right)}$ and ${\psi }_2\left(x\right)=\sqrt{n_2\left(x\right)}{e}^{i{\theta }_2\left(x\right)}$. The detuning $\delta$ prevents direct resonant tunneling between the two wells. The angle $\alpha$ between the two Raman beams allows to tune the mismatch wave vector $\vec{Q}={\vec{k}}_1-{\vec{k}}_2$. The phase difference of the two condensates $\theta \left(x\right)={\theta }_1\left(x\right)-{\theta }_2\left(x\right)$ has the shape of a staircase upon increasing $Q$ above a critical threshold. The length scale $l_J$ on which the phase jump of $2\pi$ takes place, is determined by the tunnel coupling $J$ and proportional to $1/\sqrt{J}$. By changing $Q$ one can modify the length between two adjacent solitons $l_Q\propto 1/Q$, see Refs. [1, 2].

Figure 2

Density and shifted relative phase profiles in the commensurate and incommensurate phase: For a fixed system size of $\overline{L}=40$ and $\overline{Q}=0.7$ the density profile (a) as well as the relative phase profile (b) are homogeneous. The microscopic parameters are $\overline{\mu }=30$ and $\overline{g}=0.6$ and are typical for atom chip experiments [10, 11, 27]. For a fixed system size of $\overline{L}=40$ and $\overline{Q}=1.9$ the density profile (c) shows periodic density modulations and the relative phase profile (d) shows a staircase structure in space where the dotted lines mark $2\pi$ jumps representing the transitions between two adjacent solitons. The solitonic structure implies a density modulation, i.e., the density ripples occur at the same positions of kinks in the phase difference profile degree of freedom. The (in)homogeneous phase is called the (in)commensurate phase in the context of the Pokrovsky-Talapov transition. We refer to Sec. 5 and Appendix for more details on this notion.

Figure 3

Density of solitons: the homogeneous phase is characterized by a uniform phase profile and the absence of solitons (extended red region); around $Q\simeq 1.8$ solitons are injected with a non-vanishing density (incommensurate phase). The dotted lines mark different sizes of the system where soliton injection is studied in Fig. 4. The microscopic parameters are the same as in Fig. 2.

Figure 4

For finite system size we observe discrete jumps in the number of solitons upon increasing $Q$. This effect results from fitting $m\simeq L/l_Q$ solitons into the system with size $L$. The plots correspond to the horizontal cuts in the phase diagram of Fig. 3 (dotted lines). Microscopic parameters are the same as in Fig. 2.

Figure 5

Dispersion relation. (Left) Dispersion relation of the “symmetric” (orange) and “antisymmetric” (blue) modes in the commensurate phase ($Q=0.9$). The dispersion relation of the antisymmetric modes as a function of momentum is quadratic at low energies with a gap $\mathrm{\Delta }=\sqrt{2J(2J+2\mu )}\simeq 11$ (here $J=1$ and $\mu =30$). (Right) In the incommensurate phase ($Q=2.1$), we observe two linearly dispersing sound modes at low energies. These two sound modes correspond to the $\text{U}\left(1\right)$ phonon (blue) and to the soliton phonon (orange). A scaling with system size of the low-energy eigenvalues is provided in Appendix.

Figure 6

Columns (1) and (2). Spatial profile of the density fluctuations determined from the Bogoliubov calculation for each mode. The two columns allow to categorize the modes within tube 1 and 2 into symmetric and antisymmetric modes. Symmetric modes are labeled in blue, while antisymmetric modes are labeled in orange. For instance, the third and fourth rows display the first excited state in the symmetric and antisymmetric sectors. By sorting the eigenvalues according to this symmetry considerations we show the dispersion relation of the symmetric (blue) and anti-symmetric (orange) modes in the right panel of Fig. 5. Column (3) shows the relation between the ground state current (light red) and the absolute value of the Bogoliubov mode functions (green). Note that the relative phase and the ground state current are peaked everywhere a soliton is formed in the relative phase profile of the two tubes; the modes in the antisymmetric sector are sensitive to the presence of solitons and are peaked at the same positions as the solitons. In the figure, we have plotted the absolute value of the Bogoliubov mode functions (green) to illustrate the influence of the modulation of the background field on the mode functions. The modes of the symmetric sectors show a milder spatial corrugation (see $m=2$). This is in contrast with the Pokrovsky-Talapov field theory where the symmetric modes are completely insensitive to soliton injection, because they do not couple to the antisymmetric modes.

Figure 7

Comparison with the Pokrovsky-Talapov model. Plot of $|{\rho }_{\text{mic.}}-{\rho }_{\text{PT}}|/{\rho }_{\text{mic.}}$, with ${\rho }_{\text{mic.}}$ the soliton density of the microscopic model studied in this work and ${\rho }_{\text{PT}}$ the soliton density of the effective PT field theory. A mismatch between ${\rho }_{\text{PT}}$ and ${\rho }_{\text{mic.}}$ indicates regions of the phase diagram, where corrections beyond the conventional PT description are present (microscopic parameters here are the same as in Fig. 2). Such corrections are more pronounced close to the critical point and vanish as the system size increases.

Figure 8

Finite $1/L$ scaling of the lowest eigenvalues of the Bogoliubov spectrum in the inhomogeneous phase (left, $Q=2.1$) and in the symmetric sector of the homogeneous phase (right, $Q=0.9$). The lowest energy eigenvalue crosses the horizontal axis around $L\simeq 60$ in both phases.