Local relaxation and light-cone-like propagation of correlations in a trapped one-dimensional Bose gas

We start by studying the case of an homogeneous gas with $V\left( z \right)=0$. We quickly recall the theoretical analysis introduced in [31] and derive the time evolution of the two-point relative-PCF which can be used to characterize the decay of coherence locally. The situation of two quasi-condensates in thermal equilibrium has been addressed in [32]. Here we discuss the relaxation of the PCF after the coherent splitting described above and explain the local equilibration observed recently in the experiment of [21].

We introduce the relative phase and relative density between the two gases through $\hat{\phi}={{\hat{\theta}}_{1}}-{{\hat{\theta}}_{2}}$ and $\hat{n}=\left( {{\hat{n}}_{1}}-{{\hat{n}}_{2}} \right)/2$, with the commutation relation $\left[ \hat{n}\left( z \right),\hat{\phi}\left( {{z}^{\prime }} \right) \right]=i\delta \left( z-{{z}^{\prime }} \right)$. This is motivated by interference experiments which give direct access to the relative phase field. The common (center-of-mass) degrees of freedom are described by the operators ${{\hat{\phi}}_{c}}=\left( {{\hat{\theta}}_{1}}+{{\hat{\theta}}_{2}} \right)/2$ and ${{\hat{n}}_{c}}={{\hat{n}}_{1}}+{{\hat{n}}_{2}}$. Inserting the definition of ${{\hat{\Psi}}_{j}}$ in equation (1) and neglecting third order terms ($\sim {{\left( {{\partial }_{z}}\hat{\theta} \right)}^{2}}\hat{n}$), we obtain a quadratic Hamiltonian for the phase and density operators. Assuming a symmetric splitting (equal density in each gas after splitting), the dynamics of the relative and the common degrees of freedom decouple, as shown in [23]. We obtain the following Hamiltonian governing the evolution of the relative phase and density operators:

$$\begin{eqnarray}&&\hat{H}=\mathop{\int }\nolimits {\rm{d}}z\left[ \frac{-{{\hbar }^{2}}{{n}_{0}}}{4m}{{({{\partial }_{z}}\hat{\phi})}^{2}}+g{{\hat{n}}^{2}} \right].\end{eqnarray} \tag{ 2 }$$

We rewrite this Hamiltonian in the more usual form of a Luttinger liquid [33]

$$\begin{eqnarray}&&\hat{H}=\frac{\hbar c}{2}\mathop{\int }\nolimits {\rm{d}}z\left[ \frac{K}{\pi }{{({{\partial }_{z}}\hat{\phi})}^{2}}+\frac{\pi }{K}{{\hat{n}}^{2}} \right],\end{eqnarray} \tag{ 3 }$$

with the speed of sound $c=\sqrt{g{{n}_{0}}/m}$, the Luttinger parameter $K=\frac{\hbar \pi }{2}\sqrt{\frac{{{n}_{0}}}{mg}}$ and ${{n}_{0}}$ the 1D peak density. The Luttinger liquid model is the long wavelength approximation of the many-body Hamiltonian equation (1) and allows to describe the phononic (long wavelength) excitations of the system. This approximation is valid for the description of typical ultracold atom experiments, where optical methods with an imaging resolution corresponding to a few times the healing length are used to probe the excitations in the system.

Considering periodic boundary conditions for a system of size $\mathcal{L}$, we expand the field operators as [31]

$$\begin{eqnarray}&&\hat{n}(z,t)=\frac{1}{\sqrt{\mathcal{L}}}\mathop{\mathop{\sum }\nolimits }\limits_{k\ne 0}{{\hat{n}}_{k}}(t){{e}^{ikz}},\quad \hat{\phi}(z,t)=\frac{1}{\sqrt{\mathcal{L}}}\mathop{\mathop{\sum }\nolimits }\limits_{k\ne 0}{{\hat{\phi}}_{k}}(t){{e}^{ikz}},\end{eqnarray} \tag{ 4 }$$

with the expansion coefficients given by:

$$\begin{eqnarray} {{\hat{n}}_{k}}\left( t \right) & = & \sqrt{\frac{{{n}_{0}}{{S}_{k}}}{2}}\left( {{\hat{b}}_{k}}\left( t \right)+\hat{b}_{-k}^{\dagger }\left( t \right) \right) \\ {{\hat{\phi}}_{k}}\left( t \right) & = & \frac{1}{i\sqrt{2{{n}_{0}}{{S}_{k}}}}\left( {{\hat{b}}_{k}}\left( t \right)-\hat{b}_{-k}^{\dagger }\left( t \right) \right). \end{eqnarray} \tag{ 5 }$$

Here $\hat{b}_{k}^{\dagger }$ and ${{\hat{b}}_{k}}$ are the creation and annihilation operators for an elementary excitation with momentum $\hbar k$ in the relative degrees of freedom ($k=p\times 2\pi /\mathcal{L}$ with p integer different than 0; note that the sum expands over both positive and negative k). The commutation relation for the expansion coefficients reads $\left[ {{\hat{n}}_{k}},\hat{\phi}_{k}^{\dagger } \right]=\left[ {{\hat{n}}_{k}},{{\hat{\phi}}_{-k}} \right]=i$. The structure factor in the phononic regime is given by

$$\begin{eqnarray}&&{{S}_{k}}=\frac{\hbar |k|}{2mc}=\frac{|k|K}{\pi {{n}_{0}}},\end{eqnarray} \tag{ 6 }$$

and is related to the $\left( {{u}_{k}},{{v}_{k}} \right)$ Bogoliubov coefficients used in previous works [30, 32] through $\sqrt{{{S}_{k}}}={{\left( {{u}_{k}}-{{v}_{k}} \right)}^{-1}}$.

In this basis, the Hamiltonian equation (3) takes the diagonal form

$$\begin{eqnarray} \hat{H} & = & \frac{\hbar c}{2}\mathop{\mathop{\sum }\nolimits }\limits_{k\ne 0}\left[ \frac{K}{\pi }{{k}^{2}}\hat{\phi}_{k}^{\dagger }{{\hat{\phi}}_{k}}+\frac{\pi }{K}\hat{n}_{k}^{\dagger }{{\hat{n}}_{k}} \right]+\frac{\hbar \pi c}{2K}\hat{n}_{0}^{\dagger }{{\hat{n}}_{0}} \\ {} & = & \mathop{\mathop{\sum }\nolimits }\limits_{k\ne 0}\hbar {{\omega }_{k}}\hat{b}_{k}^{\dagger }{{\hat{b}}_{k}}+\frac{\hbar \pi c}{2K}\hat{n}_{0}^{\dagger }{{\hat{n}}_{0}}, \\ \end{eqnarray} \tag{ 7 }$$

with ${{\omega }_{k}}=c|k|$. We have made explicit the term corresponding to the k = 0 mode which accounts for the global phase diffusion. The dynamics can thus be reduced to that of a set of uncoupled harmonic oscillators. It follows from the absence of coupling between modes with different momentum k that any momentum occupation numbers $\left\langle \hat{b}_{k}^{\dagger }{{\hat{b}}_{k}} \right\rangle$ that are initially imposed on the system will be conserved. Equation (7) therefore strikingly shows the integrability of the system.

We assume the coherent splitting process to be instantaneous (fast with respect to the interaction energy, ${{t}_{{\rm{split}}}}\ll h/\mu =2\pi {{\xi }_{h}}/c$) so that it can be described like a beam splitter in optics. Consequently the local distribution of atoms in each small region of the quasi-condensate (of size $\sim {{\xi }_{h}}=\hbar /mc$) is binomial, with the respective minimum uncertainty relative-phase distribution. In that way, the coherent splitting copies the phase fluctuations of the initial quasi-condensate into both parts of the split system and the relative density fluctuations are given by the local shot noise. In terms of elementary excitations, this means:

$$\begin{eqnarray}&&\left\langle \hat{\phi}_{k}^{\dagger }{{\hat{\phi}}_{k}} \right\rangle {{|}_{t=0}}=\frac{1}{2\xi _{n}^{2}{{n}_{0}}},\quad {{\left. \langle \hat{n}_{k}^{\dagger }{{\hat{n}}_{k}}\rangle \right|}_{t=0}}=\frac{\xi _{n}^{2}{{n}_{0}}}{2},\end{eqnarray} \tag{ 8 }$$

with $\xi _{n}^{2}$ being the number squeezing parameter (in the following we will take $\xi _{n}^{2}=1$ unless specified). A state where the fluctuations are initialized with an excess of relative-density fluctuations and a lack of relative-phase fluctuations is a non-equilibrium state far away from the thermal equilibrium state of the Hamiltonian equation (7) at a temperature T where:

$$\begin{eqnarray}&&\left\langle \hat{\phi}_{k}^{\dagger }{{\hat{\phi}}_{k}} \right\rangle {{|}_{{\rm{th}}}}=\frac{2}{{{\lambda }_{T}}{{k}^{2}}},\quad {{\left. \langle \hat{n}_{k}^{\dagger }{{\hat{n}}_{k}}\rangle \right|}_{{\rm{th}}}}=\frac{{{k}_{B}}T}{2g},\end{eqnarray} \tag{ 9 }$$

with ${{\lambda }_{T}}={{\hbar }^{2}}{{n}_{0}}/m{{k}_{B}}T$. The atom number fluctuations caused by the splitting correspond to the introduction of energy into the system due to the interactions. This energy manifests itself in additional excitations in the density quadrature (rather than in the phase quadrature) with respect to the thermal equilibrium (compare equations (8) and (9)).

In terms of the elementary excitations, the initial conditions of equation (8) lead to an approximately thermal-like form of the occupation numbers that reads

$$\begin{eqnarray}&&\langle \hat{b}_{k}^{\dagger }{{\hat{b}}_{k}}\rangle =\frac{{{k}_{B}}{{T}_{{\rm{eff}}}}}{\hbar {{\omega }_{k}}},\end{eqnarray} \tag{ 10 }$$

with the effective temperature given by ${{k}_{B}}{{T}_{{\rm{eff}}}}=\xi _{n}^{2}{{n}_{0}}g/2$. The fast splitting process thus equally distributes the energy ${{k}_{B}}{{T}_{{\rm{eff}}}}$ in the different modes of the system [23].

The equations of motions for the operators ${{\hat{\phi}}_{k}}\left( t \right)$ and ${{\hat{n}}_{k}}\left( t \right)$ read

$$\begin{eqnarray} i\hbar \frac{{\rm{d}}{{\hat{\phi}}_{k}}}{{\rm{d}}t} & = & \left[ {{\hat{\phi}}_{k}},\hat{H} \right]=\frac{\hbar c}{2}\frac{\pi }{K}\left( -2i{{\hat{n}}_{k}} \right), \\ i\hbar \frac{{\rm{d}}{{\hat{n}}_{k}}}{{\rm{d}}t} & = & \left[ {{\hat{n}}_{k}},\hat{H} \right]=\frac{\hbar c}{2}\frac{K}{\pi }{{k}^{2}}\left( 2i{{\hat{\phi}}_{k}} \right), \\ \end{eqnarray} \tag{ 11 }$$

and lead to harmonic oscillator equations for ${{\hat{n}}_{k}}$ and ${{\hat{\phi}}_{k}}$, with the angular oscillation frequency ${{\omega }_{k}}=ck$. With the initial conditions equation (8) and neglecting the initial phase fluctuations ($\left\langle \hat{\phi}_{k}^{\dagger }{{\hat{\phi}}_{k}} \right\rangle {{|}_{t=0}}\approx 0$, which is valid as soon as $t>h/\mu$), we obtain the evolution of the expansion coefficients:

$$\begin{eqnarray} {{\hat{\phi}}_{k}}\left( t \right) & = & \frac{\pi }{kK}\sqrt{\frac{\xi _{n}^{2}{{n}_{0}}}{2}}{\rm sin} {{\omega }_{k}}t \\ {{\hat{n}}_{k}}\left( t \right) & = & \sqrt{\frac{\xi _{n}^{2}{{n}_{0}}}{2}}{\rm cos} {{\omega }_{k}}t. \\ \end{eqnarray} \tag{ 12 }$$

These equations show the oscillation between the density and phase quadratures for the phononic excitations.

To study the decay of coherence between the two quasi-condensates and the evolution of correlations in the system, we consider the two point relative-PCF defined as [31]:

$$\begin{eqnarray} C(z,z\prime ,t) & = & \frac{\langle {{\hat{\Psi}}_{1}}(z)\hat{\Psi}_{2}^{\dagger }(z)\hat{\Psi}_{1}^{\dagger }({{z}^{\prime }}){{\hat{\Psi}}_{2}}({{z}^{\prime }})\rangle }{\langle |{{\hat{\Psi}}_{1}}(z){{|}^{2}}\rangle \langle |{{\hat{\Psi}}_{2}}({{z}^{\prime }}){{|}^{2}}\rangle } \\ {} & \simeq & {\rm exp} \left( -\frac{1}{2}\langle \Delta {{\phi }_{zz\prime }}{{\left( t \right)}^{2}}\rangle \right). \\ \end{eqnarray} \tag{ 13 }$$

In the last step we have used the fact that the fluctuations are Gaussian, which is a consequence of the quadratic Hamiltonian equation (3). Here, $\left\langle \Delta {{\phi }_{z{{z}^{\prime }}}}{{\left( t \right)}^{2}} \right\rangle \equiv \left\langle {{\left( \hat{\phi}\left( z,t \right)-\hat{\phi}\left( z\prime ,t \right) \right)}^{2}} \right\rangle$ denotes the phase variance between two points z and ${\rm{z}}\prime$ of the relative phase field. Using equation (12), the time evolution of the phase variance is given by

$$\begin{eqnarray}&&\left\langle \Delta {{\phi }_{z{{z}^{\prime }}}}{{\left( t \right)}^{2}} \right\rangle =\frac{{{\pi }^{2}}{{n}_{0}}}{\mathcal{L}{{K}^{2}}}\mathop{\mathop{\sum }\nolimits }\limits_{k\ne 0}\frac{{\rm sin} {{\left( {{\omega }_{k}}t \right)}^{2}}}{{{k}^{2}}}\left( 1-{\rm cos} \left( k\bar{z} \right) \right).\end{eqnarray} \tag{ 14 }$$

The first term in the sum equation (14) represents the growth and subsequent oscillations in the amplitude of the phase fluctuations as they get converted from the initial density fluctuations. The factor $1/{{k}^{2}}$ in the amplitude reflects the $1/k$ scaling of the excitation occupation numbers associated with the equipartition of energy induced by the fast splitting, equation (10). The term in parenthesis in the sum corresponds to the spatial fluctuations.

We will discuss the physical interpretation of this expression in more details in the next two paragraphs, considering first the dephased (prethermalized) state and second the dynamics governing the emergence of this state.

We first focus on the dephased state of the system where the two-point relative-phase variance can be approximated by its long time limit, taking ${{{\rm sin} }^{2}}\left( {{\omega }_{k}}t \right)=1/2$. Approximating the sum by an integral, we obtain

$$\begin{eqnarray}&&\langle \overline{\Delta \phi _{zz\prime }^{2}}\rangle =\frac{4mg}{{{\hbar }^{2}}}\times \frac{1}{2}\times \frac{|\bar{z}|}{2},\end{eqnarray} \tag{ 15 }$$

where we have used $\mathop{\int }\nolimits \frac{{\rm{d}}k}{2\pi }\frac{1-{\rm cos} \left( k\bar{z} \right)}{{{k}^{2}}}=\frac{|\bar{z}|}{2}$. In the relaxed (dephased) state, the PCF is thus given by

$$\begin{eqnarray}&&C(\bar{z})={\rm exp} \left( -\frac{|\bar{z}|}{{{l}_{0}}} \right),\quad {\rm{with}}\;{{l}_{0}}=\frac{2{{\hbar }^{2}}}{mg}.\end{eqnarray} \tag{ 16 }$$

The correlation length ${{l}_{0}}$ does not depend on the detail of the system (such as the density) but only on the 1D coupling constant, and is in that sense universal. The scaling comes from the equipartition of energy between the modes during the splitting.

It is interesting to discuss the role of the correlation length ${{l}_{0}}$ in the context of dephasing, linking our study with previous results obtained on phase diffusion in 3D BEC [34–36] and quasi-condensates [22]. The decay of the coherence factor $\Psi \left( t \right)\equiv \left\langle {{e}^{i\hat{\phi}\left( z,t \right)}} \right\rangle ={{e}^{-\frac{1}{2}\left\langle \hat{\phi}{{\left( z,t \right)}^{2}} \right\rangle }}$ due to the lowest mode (k = 0 term in equation (7)) is given by:

$$\begin{eqnarray}&&\Psi (t){{|}_{k=0}}\propto {{e}^{-{{t}^{2}}/\tau _{0}^{2}}},\end{eqnarray} \tag{ 17 }$$

with the characteristic time ${{\tau }_{0}}$ known from the phase diffusion in 3D BEC [34–36]: ${{\tau }_{0}}=\frac{\hbar }{g}\sqrt{\frac{L}{{{n}_{0}}}}$. On the other hand, the decay of coherence due to the contribution of all other ($k\ne 0$) modes is given by [22]

$$\begin{eqnarray}&&\Psi (t){{|}_{k\ne 0}}\propto {{e}^{-t/\tau }},\end{eqnarray} \tag{ 18 }$$

with the dephasing time $\tau =8\;{{K}^{2}}/{{\pi }^{2}}{{n}_{0}}c$ (note that our definition of K differs by $1/2$ with respect to [22]). The dephasing will therefore be dominated by 1D effects as soon as $\tau <{{\tau }_{0}}$, which correspond to the condition

$$\begin{eqnarray}&&{{l}_{0}}=\frac{1}{\xi _{n}^{2}}\frac{2{{\hbar }^{2}}}{mg}<\frac{L}{2},\end{eqnarray} \tag{ 19 }$$

where we explicitly introduced the squeezing parameter.

Recent studies on the role of atom–atom interactions in interferometers indicate that low-dimensional geometries could be favourable for interferometry [37]. However, 1D effects will limit this coherence time if the prethermalized correlation length is smaller than half the system size, whereas they can be neglected with respect to the 3D phase diffusion if ${{l}_{0}}>L/2$. The latter is, for example, the case in experiments that are working with low atom numbers, high number squeezed states or not too elongated traps [38, 39]. Equation (19) can be re-formulated in terms of experimentally controllable parameters, by expressing the amount of number squeezing required for a beam splitter so that the interferometer will not be limited by multimode dephasing:

$$\begin{eqnarray}&&\xi _{n}^{2}<\xi _{n}^{2}{{|}_{{\rm lim} }}=\frac{2\hbar }{m{{\omega }_{\bot }}{{a}_{s}}L}.\end{eqnarray} \tag{ 20 }$$

We used $g=2\hbar {{\omega }_{\bot }}{{a}_{s}}$ (in the quasi-1D regime) to express the number squeezing parameter in terms of measurable quantities (radial trap frequency ${{\omega }_{\bot }}$ and system length L). Equation (20) provides a criterion for evaluating the possible sensitivity of elongated condensates working close to the 1D regime for applications in interferometry. We illustrate this criterion in figure 1 where we represent the required number squeezing (in dB) for an interferometer to be limited by 3D phase diffusion rather than 1D effects, as a function of the physical parameters $\left( {{\omega }_{\bot }},L \right)$ controlling the 1D-ness of the system. We finally note that in the homogeneous case, condition equation (20) does not involve the linear density of the system as the dephasing time due to the k = 0 mode and that due to the $k\ne 0$ modes have the same dependence $\propto \;n_{0}^{-1/2}$. In practical applications, however, the system size and the peak density of the trapped atoms are mutually related, thus leading to an implicit density dependence of the condition equation (20).

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We now discuss the full expression equation (14) which characterizes the decay of coherence as a result of the superposition of many phononic modes. Mathematically, expression equation (14) is the fourier decomposition of a trapezoid with a siding edge at ${{\bar{z}}_{c}}=2ct$, so that the phase variance (and thus the PCF) exhibits a two step feature. This two-step feature can physically be interpreted as follows: for a given time t, short wavelength modes will grow in amplitude and linearly increase the phase variance up to a distance ${{\bar{z}}_{c}}=2ct$. Beyond that point the growth in amplitude of longer wavelength modes with $2\pi /k>{{\bar{z}}_{c}}$ exactly compensates the decrease in amplitude of the shorter wavelength modes with $2\pi /k<{{\bar{z}}_{c}}$, leading to a constant phase variance. This is illustrated in figure 2.

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Quantitatively, computing the derivative of the phase variance with respect to time, we find

$$\begin{eqnarray}&&\frac{\partial \left\langle \Delta {{\phi }_{z{{z}^{\prime }}}}{{\left( t \right)}^{2}} \right\rangle }{\partial t}=\frac{2c}{{{l}_{0}}}\times \Theta \left( 2ct-\bar{z} \right),\end{eqnarray} \tag{ 21 }$$

with $\Theta \left( x \right)$ being the Heaviside step function ($\Theta \left( x \right)=1$ if $x>0$ and 0 otherwise) and ${{l}_{0}}=2{{\hbar }^{2}}/mg$ the prethermalized correlation length introduced in equation (19). The phase randomizes with a constant rate up to the point where $\bar{z}=2ct$. Beyond that point, long-range phase coherence is retained. The full time evolution of the phase variance and its derivative, and the experimentally probed PCF is shown in figure 3.

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The form of the PCF reveals the physical interpretation of equation (21): correlations in the system take the final relaxed form (exponential decay with correlation length ${{l}_{0}}$) within the propagation cone, while long-range order remains outside of the cone. The separation $2ct$ corresponds to the distance traveled by quasi-particles moving with a velocity c in opposite directions. These quasi-particles establish the correlations between points z and ${{z}^{\prime }}=\bar{z}+z$ when they meet at a time $t=\bar{z}/2c$ [40]. Such light-cone effects have been found in various quantum many-body systems [41–45], and the local relaxation picture was first introduced in a general way in the context of the Bose–Hubbard model [46]. Here, we provide a connection to the relaxation of a Luttinger liquid, finding a similar result as the one obtained numerically for the Bose–Hubbard model [46].

We finally note that while the squeezing parameter $\xi _{n}^{2}$ influences the final relaxed state (influence on the correlation length ${{l}_{0}}$), it does not modify the velocity of correlations, i.e. the time after which the relaxed state is reached.

We now investigate the dynamics of coherence between the two gases in the case of a longitudinal trapping potential $V\left( z \right)=\frac{1}{2}m{{\omega }^{2}}{{z}^{2}}$ (see equation (1)). Under the same approximations as for the homogeneous case (neglecting third order terms in the fluctuations and considering only phononic modes), the Hamiltonian for the relative degrees of freedom can be written as

$$\begin{eqnarray}&&\hat{H}=\mathop{\int }\nolimits dz\left[ \frac{-{{\hbar }^{2}}{{n}_{0}}(z)}{4m}{{\left( {{\partial }_{z}}\hat{\phi} \right)}^{2}}+g{{\hat{n}}^{2}} \right],\end{eqnarray} \tag{ 22 }$$

where the density profile ${{n}_{0}}\left( z \right)$ solves the Gross–Pitaevski equation

$$\begin{eqnarray}&&\left[ \frac{-{{\hbar }^{2}}}{2m}\Delta +(V(z)-\mu )+g{{n}_{0}}(z) \right]\sqrt{{{n}_{0}}(z)}=0.\end{eqnarray} \tag{ 23 }$$

The Hamiltonian thus takes the form of a non-homogeneous Luttinger Liquid with spatially dependent velocity $c\left( z \right)=\sqrt{{{n}_{0}}\left( z \right)g/m}$ and Luttinger parameter $K\left( z \right)=\frac{\hbar \pi }{2}\sqrt{\frac{{{n}_{0}}\left( z \right)}{mg}}$.

Following the notations of Petrov et al [47], this Hamiltonian can be re-written as

$$\begin{eqnarray}&&\hat{H}=\frac{\hbar }{2\pi }\mathop{\int }\nolimits {\rm{d}}z\left[ {{v}_{N}}{{(\pi \hat{n})}^{2}}+{{v}_{J}}{{({{\partial }_{z}}\hat{\phi})}^{2}} \right],\end{eqnarray} \tag{ 24 }$$

with the density and phase velocities given by ${{v}_{N}}=c/K$ and ${{v}_{J}}=cK$.

For a weakly interacting gas in the Thomas–Fermi regime, the density profile is an inverted parabola

$$\begin{eqnarray}&&{{n}_{0}}(z)={{n}_{0}}\left( 1-\frac{{{z}^{2}}}{{{R}^{2}}} \right),\end{eqnarray} \tag{ 25 }$$

with the Thomas–Fermi radius $R=\sqrt{2}{{c}_{0}}/\omega$ given by peak density of the gas via ${{c}_{0}}=\sqrt{g{{n}_{0}}/m}$ and the peak density determined from the atom number [27]. In the Thomas–Fermi regime, the velocity ${{v}_{N}}$ reduces to ${{v}_{N}}=2g/\pi \hbar$ and is therefore independent on the spatial coordinate.

Proceeding as for the homogeneous system, we expand the relative-phase and relative-density fluctuations as

$$\begin{eqnarray}&&\hat{\phi}(z,t)=\mathop{\mathop{\sum }\nolimits }\limits_{j>0}{{\hat{\phi}}_{j}}(t){{f}_{j}}(z),\quad \hat{n}(z,t)=\mathop{\mathop{\sum }\nolimits }\limits_{j>0}{{\hat{n}}_{j}}(t){{f}_{j}}(z).\end{eqnarray} \tag{ 26 }$$

The equations of motion read

$$\begin{eqnarray} {{\partial }_{t}}\hat{\phi}\left( z,t \right) & = & -\pi {{v}_{N}}\hat{n}\left( z,t \right) \\ \pi {{\partial }_{t}}\hat{n}\left( z,t \right) & = & -{{\partial }_{z}}\left[ {{v}_{J}}\left( z \right){{\partial }_{z}}\hat{\phi}\left( z,t \right) \right]. \\ \end{eqnarray} \tag{ 27 }$$

Inserting the expansion equation (26) and using the expression of ${{n}_{0}}\left( z \right)$ in the Thomas–Fermi regime, we find that the eigenfunctions of the problem satisfy the equation

$$\begin{eqnarray}&&(1-{{x}^{2}})f_{j}^{\prime\prime }(x)-2xf_{j}^{\prime }(x)+\frac{2\omega _{j}^{2}}{{{\omega }^{2}}}{{f}_{j}}(x)=0,\end{eqnarray} \tag{ 28 }$$

where x = z/R and $f_{j}^{\prime }$ (resp. $f_{j}^{\prime\prime }$) denotes the first (resp. second) spatial derivative of ${{f}_{j}}$ with respect to x. The solutions are well known and can be expressed in terms of Legendre polynomials ${{P}_{j}}$:

$$\begin{eqnarray}&&{{f}_{j}}(z)=\sqrt{j+\frac{1}{2}}{{P}_{j}}(x)\;{\rm{with}}\;x=z/R,\quad {{\omega }_{j}}=\omega \sqrt{j(j+1)/2}.\end{eqnarray} \tag{ 29 }$$

The initial conditions for the mode occupation numbers after splitting for the trapped system are a priori different than for the homogeneous system (equation (8)). This is due to the fact that the amount of noise is proportional to the local density of atoms and is thus lower at the edges of the cloud than in the center. In the quantum optics picture, this would be the analogue of a local beam splitter. More specifically, the fluctuations are given by

$$\begin{eqnarray} \left\langle \hat{n}\left( z \right)\hat{n}\left( {{z}^{\prime }} \right) \right\rangle {{|}_{t=0}} & = & \frac{\xi _{n}^{2}\left( z \right){{n}_{0}}\left( z \right)}{2}\delta \left( z-{{z}^{\prime }} \right) \\ \left\langle \hat{\phi}\left( z \right)\hat{\phi}\left( {{z}^{\prime }} \right) \right\rangle {{|}_{t=0}} & = & \frac{1}{2\xi _{n}^{2}\left( z \right){{n}_{0}}\left( z \right)}\delta \left( z-{{z}^{\prime }} \right), \\ \end{eqnarray} \tag{ 30 }$$

where we introduced the local squeezing factor $\xi _{n}^{2}\left( z \right)$. It can be shown that the corresponding mode occupation numbers $\left\langle \hat{n}_{j}^{\dagger }{{\hat{n}}_{j}} \right\rangle {{|}_{t=0}}$ for the trapped system are close to that for the homogeneous system (see equation (8)) and only differ slightly for the first two modes j = 1, 2 [48]. As this small difference does not influence the final result, we will assume for clarity the same occupation for all modes, with initial density fluctuations given by

$$\begin{eqnarray}&&\langle \hat{n}_{j}^{\dagger }{{\hat{n}}_{j}}\rangle {{|}_{t=0}}=\frac{{{n}_{0}}}{2R}.\end{eqnarray} \tag{ 31 }$$

With this choice of initial conditions, the dephased state has the same effective temperature ${{T}_{{\rm{eff}}}}={{n}_{0}}g/2{{k}_{B}}$ as for the homogeneous system. These initial conditions correspond to mode occupation numbers $\left\langle \hat{b}_{j}^{\dagger }{{\hat{b}}_{j}} \right\rangle ={{k}_{B}}{{T}_{{\rm{eff}}}}/\hbar {{\omega }_{j}}$ which are the same as in thermal equilibrium [47] but with an effective temperature ${{T}_{{\rm{eff}}}}$.

Finally, combining equations (29) and (31) we obtain for the two-point phase variance:

$$\begin{eqnarray}&&\langle \Delta {{\phi }_{z{{z}^{\prime }}}}{{(t)}^{2}}\rangle =\frac{{{n}_{0}}{{\pi }^{2}}v_{N}^{2}}{2R}\sum^{\infty }_{j=1}\frac{{\rm sin} {{({{\omega }_{j}}t)}^{2}}}{\omega _{j}^{2}}{{\left[ {{f}_{j}}(z)-{{f}_{j}}(z\prime ) \right]}^{2}}.\end{eqnarray} \tag{ 32 }$$

The expression has a similar form as the one found for the homogeneous system (equation (14)). Again, the time dependent term denotes the oscillation of energy between density and phase fluctuations, and the position-dependent term reflects the spatial phase fluctuations.

The comparison of the two-point relative-PCF for the trapped and homogeneous system is shown in figure 4, for the same value of the peak density ${{n}_{0}}$. The general behaviour is similar to the homogeneous system, with a correlation function being exponential up to a characteristic distance beyond which partial long-range order remains. However, three differences appear between the two systems. First, the effective correlation length ${{\lambda }_{{\rm{eff}}}}$ corresponding to the $1/e$ value of $C\left( {\bar z} \right)$ is slightly lower than in the homogeneous case (see equation (19)), due to the presence of the Legendre polynomials in the sum equation (32). Second, the long-range order plateaus appear at slightly shorter separations because the effective correlation length is lower, and because of a smaller velocity of correlations, as we will see in the next paragraph. Finally, the correlation function drops to zero at the edge of the cloud corresponding here to $R\approx 56\ {\rm{micron}}$.

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In the homogeneous case, the light-cone-like dynamics of correlations directly comes from the Lorentz invariant form of the equation for the relative phase field. Such a light-cone condition is not obvious for the trapped system, where the speed of sound depends on the local density in the gas. Numerically computing the second derivative of the phase variance from equation (32), we find a well-defined front of correlations, the position of which linearly evolves in time. From this, we determine the characteristic velocity by extracting the position of the correlation front for various evolution times. The procedure is illustrated for three different atom numbers (peak densities) in figure 5.

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For evolution times corresponding to a distance travelled by the quasi-particles approaching the edge of the cloud at $\pm R$, the dynamics is dominated by the finite size effect (bending of the correlation function, see figure 4) and the position of the front slightly deviates from the linear scaling in time. This is illustrated by indicating the separation corresponding to half the Thomas–Fermi radius in figure 5 (horizontal lines). However, exploring this region experimentally would be very hard as it would require a very high number of realizations in order to obtain a smooth correlation function in that range of separations (more than ${{10}^{4}}$ realizations typically [21]).

In typical experiments, the gases are rather in the quasi-1D regime than perfectly 1D (i.e. $\mu ,\ {{k}_{B}}T\leqslant \hbar {{\omega }_{\bot }}$). The effect of the radial extension of the cloud can be integrated out, leading to a density profile in the longitudinal direction that is modified with respect to the Thomas–Fermi regime [49]. Treating the density profile as an inverted parabola with a modified peak density and an effective radius, we can use equation (32) to compute the dynamics of phase fluctuations and determine the position of the front of correlations numerically in the quasi-1D regime. The results are presented in figure 6 and show a lower value for the speed of correlations ($\sim 10\%$) for the same atom number, due to the slightly lower value of the radius ($\sim 4\%$) in the quasi-1D regime than in the purely 1D Thomas–Fermi regime. We emphasize that the lower velocity found for the quasi-1D system than for the Thomas–Fermi cloud is due to the more pronounced finite size effect captured in the Legendre polynomials ${{P}_{j}}\left( z/R \right)$ in equation (32), and not by the change in density between the two regimes (the latter would rather lead to a higher velocity of correlations in the quasi-1D regime because ${{n}_{0}}{{|}_{{\rm{quasi}}-1{\rm{D}}}}>{{n}_{0}}{{|}_{{\rm{TF}}}}$ by about 10%.) Therefore, this difference could not have been captured by a homogeneous system calculation, which would have overestimated the velocity of correlations. Our trapped calculation for the quasi-1D regime is in good agreement with experiments [21].

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