Geometry-induced modification of fluctuation spectrum in quasi-two-dimensional condensates

We employ HFB-Popov approximation [29–37] to examine the properties of the collective excitations at T = 0, and impact of U₀ on the variation in the density distribution at $T\ne 0$. In this formalism, the Bose field operator $\hat{{\rm{\Psi }}}$ is decomposed into two parts; the c-field or the condensate part represented by $\phi (x,y,t)$ and the non-condensate or the fluctuation part denoted by $\widetilde{\psi }(x,y,t)$. With $\hat{{\rm{\Psi }}}=\phi +\widetilde{\psi }$, in the mean-field domain the generalized GP equation with the time-independent HFB-Popov approximation is

$$\begin{eqnarray}&&\hat{h}\phi +U[{n}_{{\rm{c}}}+2\tilde{n}]\phi =0,\end{eqnarray} \tag{ 2 }$$

where, $\hat{h}=(-{{\hslash }}^{2}/2m)({\partial }^{2}/\partial {x}^{2}+{\partial }^{2}/\partial {y}^{2})+V(x,y)-\mu$ represents the single-particle part of the Hamiltonian; ${n}_{{\rm{c}}}(x,y)\equiv | \phi (x,y){| }^{2}$, $\tilde{n}(x,y)\equiv \langle {\widetilde{\psi }}^{\dagger }(x,y,t)\widetilde{\psi }(x,y,t)\rangle$, and $n(x,y)={n}_{{\rm{c}}}(x,y)+\tilde{n}(x,y)$ are the local condensate, non-condensate, and total density, respectively. Applying Bogoliubov transformation, $\widetilde{\psi }(x,y,t)$ in terms of quasiparticle modes is

$$\begin{eqnarray}&&\widetilde{\psi }=\displaystyle \sum _{j}[{u}_{j}(x,y){\hat{\alpha }}_{j}(x,y){{\rm{e}}}^{-{\rm{i}}{E}_{j}t/{\hslash }}-{v}_{j}^{* }(x,y){\hat{\alpha }}_{j}^{\dagger }(x,y){{\rm{e}}}^{{\rm{i}}{E}_{j}t/{\hslash }}].\ \ \ \ \ \end{eqnarray} \tag{ 3 }$$

Here, ${\hat{\alpha }}_{j}$ (${\hat{\alpha }}_{j}^{\dagger }$) are the quasiparticle annihilation (creation) operators and satisfy the usual Bose commutation relations, and the subscript j represents the energy eigenvalue index. The functions u_j and v_j are the Bogoliubov quasiparticle amplitudes corresponding to the jth energy eigenstate. Using the above ansatz and definitions we arrive at the following Bogoliubov–de Gennes equations

$$\begin{eqnarray}(\hat{h}+2{Un}){u}_{j}-U{\phi }^{2}{v}_{j} & = & {E}_{j}{u}_{j},\\ -(\hat{h}+2{Un}){v}_{j}+U{\phi }^{* 2}{u}_{j} & = & {E}_{j}{v}_{j}.\end{eqnarray} \tag{ 4 }$$

The thermal or non-condensate density $\tilde{n}$ at temperature T is

$$\begin{eqnarray}&&\tilde{n}=\displaystyle \sum _{j}\{[| {u}_{j}{| }^{2}+| {v}_{j}{| }^{2}]{N}_{0}({E}_{j})+| {v}_{j}{| }^{2}\},\end{eqnarray} \tag{ 5 }$$

where, $\langle {\hat{\alpha }}_{j}^{\dagger }{\hat{\alpha }}_{j}\rangle ={({{\rm{e}}}^{\beta {E}_{j}}-1)}^{-1}\equiv {N}_{0}({E}_{j})$ with $\beta =1/{k}_{{\rm{B}}}T$, is the Bose factor of the jth quasiparticle state with energy E_j at temperature T. However, it should be emphasized that, when $T\to 0$, ${N}_{0}({E}_{j})$'s in equation (5) vanish. The non-condensate density is then reduced to $\tilde{n}={\sum }_{j}| {v}_{j}{| }^{2}$ which represent quantum fluctuations.

It must be mentioned here that the HFB-Popov approximation is gapless, and satisfies Hugenholtz–Pines theorem [38]. Using the theory it is possible to calculate the quasiparticle excitations, and obtain the equilibrium density profiles of the condensate and the thermal densities. The essence of the HFB-Popov theory is to obtain stationary states as self-consistent solutions of equations (2) and (4). However, the omission of the anomalous average, which accounts for the transfer of particles between condensate and non-condensate clouds, in this theory leads to non-conservation of particle number. This is in contrast to number-conserving theories where the increase in the number of condensate particles should imply a decrease in the number of non-condensate particles. In the present work, through normalization, we maintain constant number of particles, and this translates to a change in the chemical potential.

The time-dependent HFB-Popov theory is well suited to study the quasiparticle modes, and the quantum and thermal fluctuations at equilibrium. It is, however, not suitable to the study the dynamics of condensates [39]. The SGPE, on the other hand, provides a good description of the dynamical evolution of the condensate, and non-condensate clouds. In fact, the experimental realizations have close correspondence to the SGPE results, where the individual results obtained with independent noise realizations are representatives of independent experimental observations. Hence, as one may expect, the experimental results for a system at equilibrium are equivalent to the ensemble average of results from SGPE [40–42]. For the present work we employ SGPE to examine the realistic density distribution of the condensate and non-condensate clouds. This provides an analysis of the thermal fluctuations without the knowledge of quasiparticle spectrum, and a comparison with the HFB-Popov results has the potential to understand the effects of fluctuations. In the present work we use a scheme based on solving the stochastic Langevin equation for BEC [43] in toroidal traps, and the SGPE is [42]

$$\begin{eqnarray}&&{\rm{i}}{\hslash }\displaystyle \frac{\partial {\rm{\Psi }}({\bf{x}},t)}{\partial t}=(1-{\rm{i}}\gamma )\left[-\displaystyle \frac{{{\hslash }}^{2}{{\rm{\nabla }}}^{2}}{2m}+{V}_{{\rm{net}}}+U| {\rm{\Psi }}({\bf{x}},t){| }^{2}-\mu \right]{\rm{\Psi }}({\bf{x}},t)+\eta ({\bf{x}},t).\end{eqnarray} \tag{ 6 }$$

The constant γ denotes the dissipation, and μ is the chemical potential. Following the fluctuation–dissipation theorem required for equilibration, the random fluctuation $\eta ({\bf{x}},t)$ satisfies the following noise–noise correlation

$$\begin{eqnarray}&&\langle \eta ({\bf{x}},t){\eta }^{* }({\bf{x}}^{\prime} ,t^{\prime} )\rangle =2\gamma {k}_{{\rm{B}}}T{\hslash }\delta ({\bf{x}}-{\bf{x}}^{\prime} )\delta (t-t^{\prime} ),\end{eqnarray} \tag{ 7 }$$

where k_B and T are the Boltzmann constant and temperature respectively, and $\langle \ldots \rangle$ denotes the ensemble average, where each member represents realizations with difference arising from fluctuations. The essence of SGPE is to divide the system into two parts. In which a few highly occupied low-energy quasiparticles are subsumed in the Langevin field ${\rm{\Psi }}({\bf{x}},t)$, and the other part represents a heat bath denoted by the noise term η. The latter incorporates the effects of the higher energy quasiparticles. Another description of the Langevin field is that it is the expectation of the field operator $\hat{{\rm{\Psi }}}$ at zero temperature, and at finite temperatures has contributions from the thermal fluctuations.

In the SGP formalism, the physical properties are computed as an ensemble average over several independent noise realizations, and the total number of atoms at equilibrium is fixed by the chemical potential μ irrespective of the change in temperature [44]. As an example the density profile or $| {\rm{\Psi }}({\bf{x}},t){| }^{2}$ of one realization is shown in figure 1. From the density, at equilibrium, the quasicondensate density ${n}_{{\rm{qc}}}(x)$ is extracted through the second order correlation [45–47] as

$$\begin{eqnarray}&&{n}_{{\rm{qc}}}(x)=n(x)\sqrt{2-{g}^{(2)}(x)},\end{eqnarray} \tag{ 8 }$$

where $n(x)=| {\rm{\Psi }}(x){| }^{2}$, and ${g}^{(2)}(x)=\langle | {\rm{\Psi }}(x){| }^{4}\rangle /\langle | {\rm{\Psi }}(x){| }^{2}{\rangle }^{2}$ is the second order correlation function. Thus the quantity $n(x)-{n}_{{\rm{qc}}}(x)$ gives us the non-condensate or the thermal density. Pertinent to the present work and as mentioned earlier, thermal fluctuations in toroidal potentials have a direct bearing on the KZ mechanism as discussed in [48, 49]. These previous theoretical investigations, however, regard the toroidal systems as quasi-1D. In the present work we examine the validity of this assumption in the context of finite temperatures effects, and the observations based on the results from HFB-Popov and SGP theories are given in the next section.

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In the absence of the Gaussian potential or when ${U}_{0}=0$, the profile of n_c has rotational symmetry with the maximum located at the origin, and decreases to zero with $r\gt 0$. The corresponding quasiparticle spectrum has a Goldstone mode, and doubly degenerate Kohn modes with $\omega /{\omega }_{\perp }=1$. The quasiparticle amplitudes corresponding to one of the degenerate Kohn modes is shown in figure 3(a). For ${U}_{0}\ne 0$, n_c shows a dip in the central region, and an overall increase in the radial extent due to the repulsive Gaussian potential. As U₀ is increased when ${U}_{0}\approx \mu$, which occurs when ${U}_{0}\approx 15{\hslash }{\omega }_{\perp }$, the condensate cloud is toroidal in shape and hence ${n}_{{\rm{c}}}(0,0)\to 0$. The change in the geometry, pancake to toroidal, of the n_c modifies the excitation spectrum and the structure of the Bogoliubov quasiparticle amplitudes. An important observation associated with the change is, the wavelength of excitations becomes longer as they now lie along the circumference of the toroid. This decreases the quasiparticle energies, and the evolution of mode energies as a function of U₀ is shown in figure 2. From the figure, the Kohn mode getting soft with increase in U₀ is discernible. The metamorphosis of the Kohn mode amplitude from pancake to toroidal geometry of the BEC is shown in figure 3. From the mode functions the ratio of the Kohn mode wavelength at ${U}_{0}=20{\hslash }{\omega }_{\perp }$ and ${U}_{0}=0$ is $\approx 4.0$, which is close to π, the ratio between the diameter and circumference of a circle. It deserves to be mentioned here that for $15{\hslash }{\omega }_{\perp }\lt {U}_{0}\lt 20{\hslash }{\omega }_{\perp }$, the energy of the Kohn mode gets saturated. This is because the variation in the circumference of the condensate is small, and the change in the wavelength of excitations is negligible.

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One distinct trend in the evolution of selected neighbouring mode energy pairs is discernible from the figure. As an example, consider the evolution of breathing and hexapole modes with $\omega /{\omega }_{\perp }=2.00$, and 2.06, respectively, at ${U}_{0}=0$. As U₀ is increased, in the domain $0\lt {U}_{0}\lt 7.5{\hslash }{\omega }_{\perp }$ energies of these two modes are lowered with decreasing separation. At ${U}_{0}\approx \lt 7.5{\hslash }{\omega }_{\perp }$ the two modes cross over, and when ${U}_{0}\gt 7.5{\hslash }{\omega }_{\perp }$ the breathing mode energy increases, however, the hexapole mode energy continues to decrease. The reason behind the increase in breathing mode energy is because it is a l = 0 mode it remains a radial excitation. So, when ${U}_{0}\gt 7.5{\hslash }{\omega }_{\perp }$ as the trapping potential assumes toroidal geometry, the effective radial frequency is $\approx 1.63{\omega }_{\perp }$. The hexapole mode, which has l = 3 is transformed to one along axis of the toroid, and hence, decreases in energy. Similar trend occurs in other neighbouring pairs of states with l = 0, and the other with $l\gt 0$. It is to be mentioned that, the modes cross over. These crossing are not avoided because l is a good quantum number, and quasiparticles with different l do not mix.

At zero temperature the quantum fluctuations, which depend on the $| v{| }^{2}$ of the quasiparticle amplitudes, produce finite $\tilde{n}$. Upon closer inspection, we find that the dominant contribution to the number of non-condensate atoms $\tilde{N}=\int \tilde{n}(x,y)\,{\rm{d}}x{\rm{d}}y$ arises from the doubly degenerate Kohn mode. With the variation of U₀, there is an enhancement of $\tilde{n}$ as U₀ is increased. There are two reasons behind this, and both are associated with the transition of n_c from pancake to toroidal structure. First, the Kohn mode energy is lowered, and second, the extrema of the mode functions coincide with the density maxima of n_c.

For $T\ne 0$, the Bose factors ${N}_{0}({E}_{J})\ne 0$, so in addition to the quantum fluctuations $\tilde{n}$ have contributions from the thermal fluctuations as well. The latter, however, are orders of magnitude larger in the temperature domain of experimental interest $\approx 10$ nK. Albeit we have done computations for different temperatures, for the aforementioned reason, we examine the HFB-Popov results for T = 10 nK. The higher $\tilde{n}$ at finite temperature enhances the interaction between the condensate and non-condensate atoms, represented by the $\tilde{n}\phi$ term in equation (2), and modifies n_c. Due to the repulsive inter-atomic interactions the profile of $\tilde{n}$ develops a dip in the central region when ${U}_{0}=0$, where the maximum of n_c is located. In general, as evident from the density plots in figure 4, due to higher energy the spatial extent of the thermal cloud is larger than the condensate cloud. With higher U₀, n_c develops a dip at the center, and as mentioned in the previous section, is transformed to a toroidal condensate when ${U}_{0}\approx 15{\hslash }{\omega }_{\perp }$. The transformation is evident from the density plots of n_c as shown in figure 5. As n_c and $\tilde{n}$ are coupled, the density structure of $\tilde{n}$ is also altered. At ${U}_{0}=15{\hslash }{\omega }_{\perp }$, ${n}_{{\rm{c}}}(0,0)\approx 0.0$ where as $\tilde{n}(0,0)$ is still $\approx 6 \%$ of the maximum value. This difference is due to lower n_c in the central region, and higher energy of the non-condensate atoms. The progressive depletion of n_c from the central region with higher U₀, and the corresponding changes in $\tilde{n}$ are prominent in figures 5 (d) and (h). At lower amplitude of the Gaussian potential, ${U}_{0}\lt 15{\hslash }{\omega }_{\perp }$, the maxima of n_c and $\tilde{n}$ do not coincide and the separation is $\approx 2.0{a}_{{\rm{osc}}}$. However, when ${U}_{0}=15{\hslash }{\omega }_{\perp }$ the separation between maxima is reduced to $\approx 0.1{a}_{{\rm{osc}}}$, the trend in the shifting of the maxima is shown in figure 4 through the plot of densities along the x-axis. The number of thermal atoms increases $\approx 2.5$ fold with increasing the strength of U₀ from 0 to $15{\hslash }{\omega }_{\perp }$.

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The KZ mechanism has been the subject of intense investigations in toroidal or multiply connected condensates [48, 49, 52]. One key simplifying assumption in all the works is the consideration of a toroidal condensate as a 1D system with periodic boundary condition. A natural outcome of this assumption is, at finite temperatures the maxima of the condensate and non-condensate densities must lie along the axis of the toroid. Hence, the KZ scaling laws obtained in the previous works [48, 49, 52] are applicable in toroidal condensates only when the parameters of the toroid satisfies the conditions $\mu \ll {U}_{0}$, and this is an additional criterion to ensure the alignment of the thermal fluctuations. Otherwise, as observed in the present results, the non-condensate or fluctuations distribution have a different geometry, and the system deviates from the 1D approximation. This is an important finding which can only be obtained from finite temperature theories.