High-temperature nonequilibrium Bose condensation induced by a hot needle

TL;DR

This paper theoretically demonstrates that a one-dimensional Bose gas driven far from equilibrium by a hot localized bath can exhibit high-temperature Bose condensation, including into excited states, due to suppression of long-wavelength excitations.

Contribution

It introduces a novel nonequilibrium setup with a hot needle that induces high-temperature Bose condensation and explores the conditions favoring ground or excited state condensation.

Findings

Bose condensation occurs at temperatures much higher than equilibrium.

Condensation can shift from ground to excited states with increasing needle temperature.

Suppression of long-wavelength excitations explains the high-temperature condensation.

Abstract

We investigate theoretically a one-dimensional ideal Bose gas that is driven into a steady state far from equilibrium via the coupling to two heat baths: a global bath of temperature $T$ and a "hot needle", a bath of temperature $T_h\gg T$ with localized coupling to the system. Remarkably, this system features a crossover to finite-size Bose condensation at temperatures $T$ that are orders of magnitude larger than the equilibrium condensation temperature. This counterintuitive effect is explained by a suppression of long-wavelength excitations resulting from the competition between both baths. Moreover, for sufficiently large needle temperatures ground-state condensation is superseded by condensation into an excited state, which is favored by its weaker coupling to the hot needle. Our results suggest a general strategy for the preparation of quantum degenerate nonequilibrium steady states with unconventional properties and at large temperatures.

Equations

$$⟨{\hat{n}}_k⟩=\frac{1}{e^{({\epsilon }_k-\mu )/T}-1}$$

$$⟨{\hat{n}}_k⟩\simeq \frac{T}{{\epsilon }_k-\mu }\simeq \frac{T}{J{k}^2-2J-\mu },$$

$$\mu =-2J+J{k}_0^2-T/N_c.$$

$$N^{\prime }=\sum _{k^{\prime }\ne k_0}⟨{\hat{n}}_k⟩\simeq \sum _{\nu =2}^{\infty }\frac{1}{\frac{J{\pi }^2}{T{M}^2}({\nu }^2-1)+\frac{1}{N_c}}.$$

$$T_c^{\text{eq}}\simeq \frac{a{\pi }^2}{2}\frac{nJ}{M}\approx 8.3\frac{nJ}{M},$$

$$\dot{p}_{\bm{n}}=\sum_{kq}(1+n_{q})n_{k}\big{(}R_{kq}p_{{\bm{n}}_{q\leftarrow k}}-R_{qk}p_{\bm{n}}\big{)},$$

$$\begin{array}{rl}0& \approx A_{k{k}_0}^{(g)}⟨{\hat{n}}_k⟩N_c+R_{k{k}_0}^{(g)}{N}_c+\\ & \sum _{q\ne k_0}{A}_{kq}⟨{\hat{n}}_k⟩⟨{\hat{n}}_q⟩+\sum _{q\ne k_0}{R}_{kq}⟨{\hat{n}}_q⟩-\sum _q{R}_{qk}⟨{\hat{n}}_k⟩,\end{array}$$

$$\begin{array}{rl}0=& R_{k{k}_0}^{(g)}{N}_c+\sum _{q\ne k_0}{R}_{kq}^{(h)}⟨{\hat{n}}_q⟩-\sum _q{R}_{qk}^{(h)}⟨{\hat{n}}_k⟩\\ & +\mathcal{O}(T/T_h),\end{array}$$

$$0\approx$$

$$\approx$$

$$⟨{\hat{n}}_k⟩=\frac{N_c{\gamma }^{2}T}{M2{\gamma }_h^2{\ell }^2{T}_h{k}^2}=\frac{N_c{w}^2}{2M}\frac{1}{k^2},$$

$$N^{\prime }\approx N_c{w}^2\frac{M}{2{\pi }^2}\sum _{\nu =2}^{\infty }\frac{1}{{\nu }^2}.$$

$$T_c^{\text{ne1}}\approx 30.6\frac{{\ell }^2{\gamma }_h^2{T}_h}{{\gamma }^2}\frac{1}{M}.$$

$$\begin{array}{r}0\approx \frac{N^{\prime }}{M}{N}_c\sum _q{A}_{k_{0}q}^{(g)}+\frac{N^{\prime }}{M}\sum _{q\ne k_0}{R}_{k_{0}q}^{(g)}-N_c\sum _{q\ne k_0}{R}_{q{k}_0}^{(g)}.\end{array}$$

$$0\approx\gamma^{2}N_{c}\,\biggl{[}2JN^{\prime}-Ts(T)\biggr{]},$$

$$s(T)=\sum _{q\ne k_0}\frac{{\Delta }_{q{k}_0}/T}{e^{{\Delta }_{q{k}_0}/T}-1}\stackrel{T\gg J}{⟶}M-1.$$

$$N^{\prime }=\frac{Ts(T)}{2J}.$$

$$\langle n_{k}\rangle=\frac{\sum_{q}R_{kq}\langle n_{q}\rangle}{\sum_{q}\bigl{(}A_{qk}\langle n_{q}\rangle+R_{qk}\bigr{)}}.$$

$$A_{k_{c}k}N+\sum _q{R}_{qk}>0.$$

$${\gamma }^2{\Delta }_{k{k}_c}N+{\gamma }_h^2{T}_{h}2M\sin (k\ell {)}^2>0$$

$$T_h>T_h^{s,\alpha }=\frac{n{\gamma }^2}{2(\ell {\gamma }_h{)}^2}\underset{{\alpha }^{\prime }<\alpha }{\max }\left(\frac{{\Delta }_{k_{\alpha }{k}_{{\alpha }^{\prime }}}}{{\delta }_{{\alpha }^{\prime }}^2}\right)$$

$$\hat{H}=\int_{0}^{L}\!{\mathrm{d}}x\,\hat{\psi}^{\dagger}(x)\bigg{(}-\frac{\hbar^{2}}{2m}\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}x^{2}}\bigg{)}\hat{\psi}(x)=\sum_{k}\varepsilon_{k}\hat{n}_{k},$$

$${\hat{v}}_h=\frac{L}{d}{\int }_{\ell -\frac{d}{2}}^{\ell +\frac{d}{2}}\mathrm{d}x{\hat{\psi }}^{†}(x)\hat{\psi }(x).$$

Full text

Supplemental material for “High-temperature nonequilibrium Bose condensation induced by a hot needle”

Alexander Schnell

schnell@pks.mpg.de

Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany

  

Daniel Vorberg

Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany

  

Roland Ketzmerick

Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany

Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany

  

André Eckardt

eckardt@pks.mpg.de

Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany

In this supplemental material we (i) estimate the condensation temperature for a one-dimensional tight-binding chain of finite length $M$ in equilibrium; (ii) justify the kinetic equations of motion obtained from the meanfield approximation by comparing it to Monte-Carlo results for the exact master equation; (iii) estimate the nonequilibrium condensation temperatures for the 1D and the 3D-like transition; (iv) estimate the switch temperatures for excited state condensation; and (v) show that the nonequilibrium condensation phenomena found for a discrete tight-binding chain can also be observed in a continuous one-dimensional system.

I Condensation temperature in equilibrium

Under equilibrium conditions, when the system is coupled only to the global bath of temperature $T$ (i.e. for $\gamma_{h}=0$), Equation (4) (of the main text) is solved by the grand-canonical mean occupations

[TABLE]

with chemical potential $\mu$. When (finite-size) Bose condensation sets in, $\mu$ approaches $\varepsilon_{k_{0}}$ from below, so that the occupations of the low-energy modes with $k\ll 1$ can be approximated by

[TABLE]

where we have used $\varepsilon_{k}=-2J\cos(k)\simeq-2J+Jk^{2}$. The chemical potential can be expressed in terms of the occupation $N_{c}=\langle\hat{n}_{k_{0}}\rangle$ of the ground state with wave number $k_{0}=\pi/(M+1)$,

[TABLE]

For low temperatures, the number $N^{\prime}$ of particles occupying excited states, with $k=\nu\pi/(M+1)$, is dominated by the long-wavelength modes, so that we can approximate

[TABLE]

For a finite system, we define the characteristic temperature $T_{c}$, where Bose condensation sets in, as the temperature for which half of the particles occupy the single-particle ground state, $N^{\prime}=N_{c}=N/2$. It is given by

[TABLE]

where $a\approx 1.68$ solves $1=a\sum_{\nu=2}^{\infty}1/(\nu^{2}+a-1)$. In Fig. 1 we plot the ground-state occupation (i.e. the condensate fraction) of the tight binding chain together with the estimate (A.5) for the condensation temperature $T^{\text{eq}}_{c}$. The inverse dependence of $T^{\text{eq}}_{c}$ on the system size $M$ reflects the well-known result that in one spatial dimension, in the thermodynamic limit Bose-Einstein condensation is suppressed by thermal long-wavelength fluctuations.

II Quasiexact Monte-Carlo results for a small system

The meanfield approximation $\langle\hat{n}_{q}\hat{n}_{k}\rangle\approx\langle\hat{n}_{q}\rangle\langle\hat{n}_{k}\rangle$, which gives rise to the closed set of kinetic equations (4) for the mean occupations, allows us to treat large systems of up to $M=10^{4}$ lattice sites and to find analytical estimates for the parameters where condensation sets in. In order to justify this approximation, we have also simulated the full many-body rate equation for the probability distribution $p_{\bm{n}}$ for finding the system in the eigenstate $|{\bm{n}}\rangle$. It reads

[TABLE]

where ${\bm{n}}_{q\leftarrow k}$ denotes the vector of occupation numbers obtained from ${\bm{n}}$ by transferring a particle from mode $k$ to mode $q$.

An efficient way of solving this equation is given by quantum-jump Monte-Carlo simulations (see, e.g., Plenio and Knight (1998)). For that purpose we generate a random walk in the classical space of Fock states ${\bm{n}}$ (which is exponentially large with respect to the system size, but much smaller than the Fock space, which contains also the coherent superpositions of the Fock states). Namely, according to the sum and the relative weight of the many-body rates $R_{qk}(n_{q}+1)n_{k}$ leading away from the current state ${\bm{n}}$, we draw both the time after which a quantum jump happens and the new state ${\bm{n}}_{q\leftarrow k}$, respectively. Expectation values like the mean occupations $\langle\hat{n}_{k}\rangle$ are computed by averaging over a random path. This method gives quasiexact results, in the sense that the accuracy is controlled by the length of the random path. A detailed description of the method is given in Ref. Vorberg et al. (2015).

In Fig. 2 we plot the mean occupations $\langle\hat{n}_{k}\rangle$ of a system of $M=50$ sites and $n=3$, $\ell=3$, $\gamma_{h}/\gamma=1$, and $T_{h}=120J$. The Monte-Carlo data (red crosses) are reproduced almost perfectly by the meanfield solution (solid lines). Already in this rather small system, we can see a rather sharp crossover, to a Bose condensed regime at $T_{c}^{\text{ne}}$.

III Nonequilibrium condensation temperature

As described in the main text, in our system a Bose condensate in the ground state $k_{0}=\pi/(M+1)$ can be destroyed either by a large occupation of long-wavelength modes or by a large occupations of excited states at all energies. While the former case resembles condensation in a 1D system in equilibrium and leads to a system-size dependent condensation temperature $T_{c}^{\text{ne1}}$, the latter case bears similarity to equilibrium condensation in 3D and happens at a condensation temperature $T_{c}^{\text{ne2}}$, which is independent of the system size. Below, we will estimate both temperatures. The generalization to excited-state condensation is straightforward and not presented here.

III.1 Condensation destroyed by long-wavelength modes

Let us suppose that the system is in the condensed regime, where a large fraction $N_{c}/N$ of the particles occupy the ground-state mode and the modes at large quasimomenta are approximately unoccupied (which requires $T\ll T_{c}^{\text{ne}2}$, see next section). The mean occupations $\langle\hat{n}_{k}\rangle$ of the long-wavelength modes $k$ with $k_{0}<k\ll 1$ in the vicinity of $k_{0}$ are then obtained from

[TABLE]

which was derived from Eq. (4) by neglecting the weak residual coupling of the ground-state mode to the hot needle, $R^{(h)}_{kk_{0}}\approx 0\approx A^{(h)}_{kk_{0}}$ . The last term describes the coupling to approximately empty modes at larger quasimomenta $q$.

We can now use the assumption of a large needle temperature $T_{h}\gg T$. Employing the scaling $\langle\hat{n}_{k}\rangle\propto T/T_{h}$ (which will be shown to be self-consistent below), we find

[TABLE]

where we obtained the scaling of $R_{kq}$ and $A_{kq}$ from Eqs. (2) and (3) from the main text. The rates $R^{(h)}_{kq}$ scale like $k^{2}$ for small $k$. So for $k$ going to zero, we may omit the second term, but not the third one since $\langle\hat{n}_{k}\rangle$ is peaked around zero, which will keep the third term finite. This yields

[TABLE]

For the second approximation we used the small-$|\Delta_{kq}|/T$ expression, Eq. (2), for the rates $R^{(g)}_{kk_{0}}$ and $\sum_{q}R^{(h)}_{qk}\approx 4\gamma_{h}^{2}T_{h}\sin^{2}(\ell k)\sum_{q}\sin^{2}(\ell q)\approx 2\gamma_{h}^{2}\ell^{2}k^{2}MT_{h}$. Thus, for large $T_{h}$ we have

[TABLE]

with $w$ as defined in the main text. The occupations scale like $T/T_{h}$ as required by self consistence. Even though expression (A.11) is derived for modes $k$ with $k_{0}<k\ll 1$, we can use it for all $k\neq k_{0}$ since it vanishes rapidly with increasing $k$. With that, the condensate depletion due to the long wave modes $N^{\prime}=\sum_{k\neq k_{0}}\langle\hat{n}_{k}\rangle$ is approximately given by

[TABLE]

Using the definition that at the condensation temperature $N_{c}=N^{\prime}=N/2$, we can estimate

[TABLE]

III.2 Condensation destroyed by modes with large quasimomentum

Let us again suppose that the system is in the condensed regime with a large fraction $N_{c}/N$ occupying the ground-state mode. But now, we assume that $T\ll T_{c}^{\text{ne}1}$ so that the long-wavelengths modes are hardly occupied. Furthermore, we assume that the occupations of the most of the excited modes $k\neq k_{0}$ are close to a flat distribution corresponding to the hot temperature $T_{h}\gg J$, so that we can approximate $\langle\hat{n}_{k}\rangle\approx N^{\prime}/M$ for $k\neq k_{0}$. Plugging this ansatz into the rate equation (4) we find for $k=k_{0}$

[TABLE]

Analogous to Eq. (A.7), here we assumed that the ground state $k_{0}$ decouples from the needle. In the condensed regime we find $N_{c}\gg T/J$, and $N_{c}\gg N^{\prime}/M$ and therefore the first and the last term dominate over the second one. Using $\sum_{q}A^{(g)}_{k_{0}q}=\gamma^{2}\sum_{q}\Delta_{qk_{0}}\approx\gamma^{2}2JM$ this leads to

[TABLE]

where we have defined the sum

[TABLE]

So we find the total depletion to be given by

[TABLE]

Solving this equation for $N_{c}=N^{\prime}=nM/2$, we may obtain $T_{c}^{\text{ne2}}$ numerically. However, with the limiting value of $s(T_{c}^{\text{ne2}})\approx M-1$ from Eq. (A.16) the condensation temperature reads $T_{c}^{\text{ne2}}\approx nJ$, which is consistent with the assumption $T_{c}^{\text{ne2}}\gg J$ as long as $n\gg 1$.

IV Temperature for switch to excited state condensation

Let us now discuss, under which conditions we can expect condensation in an excited state, $k_{c}=k_{\alpha}\approx\frac{\pi}{\ell},\frac{2\pi}{\ell},\dots$, in the finite system. The mean field equation (4) must hold for all $k$, which we may solve formally like

[TABLE]

We will use the fact that all occupations must be non-negative, $\langle n_{k}\rangle\geq 0$, to find the regime where excited state condensation may occur.

Since the numerator of Eq. (A.18) is strictly positive, the denominator has to be positive, too. Deep in the condensed regime, we may neglect the depletion, $N_{c}\approx N$, which yields

[TABLE]

Considering that the condensate decouples from the hot bath, the rate imballance $A_{k_{c}k}$ is dominated by the global bath and we can approximate $A_{k_{c}k}\approx\gamma^{2}\Delta_{kk_{c}}$. Additionally $T_{h}\gg T$ holds, such that $\sum_{q}R_{qk}\approx\sum_{q}R^{(h)}_{qk}\approx\gamma_{h}^{2}T_{h}2M\sin(k\ell)^{2}$. In total, we have to satisfy

[TABLE]

for all $k$. For ground-state condensation $k_{c}=k_{0}={\pi}/({M+1})$ both terms are positive, such that for all parameter values the ground state condensate ansatz does not violate the physical requirement of positive occupations. For $k_{\alpha}\approx\kappa_{\alpha}=\frac{\alpha\pi}{\ell}$ with $\alpha\geq 1$ however, the first term is negative for all $k<k_{\alpha}$ and must be compensated by the second term. Since the second term is minimal at one of the decoupling wavenumbers $k_{\alpha^{\prime}}$, $\alpha^{\prime}<\alpha$, it suffices to check for positivity at these wave numbers. From this requirement, we find that for

[TABLE]

condensation in the excited state $k_{\alpha}$ may occur.

V Continuous system

In the main text, we treated a one-dimensional (1D) system of noninteracting bosons in a tight-binding lattice. However, the effects discussed in the main text do not depend on the discrete nature of the tight-binding lattice. They occur in a similar form also in a continuous 1D system described by the Hamiltonian

[TABLE]

where $m$ denotes the mass of the particles and $\hat{\psi}(x)$ the field operator annihilating a boson at position $x$. The dimensionless wavenumbers $k=\nu\pi d/L$ with $\nu=1,2,\ldots$ and some length scale $d$ (which we take to be the extent of the hot needle defined below) characterize the single-particle eigenstates with energy $\varepsilon_{k}=\frac{\hbar^{2}k^{2}}{2md^{2}}\equiv Ck^{2}$, and wave functions $\langle x|k\rangle=\sqrt{2/L}\,\sin(kx/d)$. The corresponding number operator reads $\hat{n}_{k}=\hat{c}^{\dagger}_{k}\hat{c}_{k}$ with $\hat{c}_{k}=\int_{0}^{L}\!{\mathrm{d}}x\,\langle k|x\rangle\,\hat{\psi(x)}$. The energy eigenstates of the system are Fock states $|{\bm{n}}\rangle$ labeled by the vector of occupation numbers $n_{k}$.

In such a continuous system, a hot local bath at position $x=\ell$ of spatial extent $d$, can be described by the coupling operator

[TABLE]

The corresponding single-particle rates are of the form of Eq. (2), with $f^{(h)}_{kq}=[\cos(k\frac{\ell}{d})\,\mathrm{sinc}(k/2)-\cos(q\frac{\ell}{d})\,\mathrm{sinc}(q/2)]^{2}$ giving $f^{(h)}_{kq}\simeq 4\sin^{2}(k{\ell}/{d})\sin^{2}(q{\ell}/{d})$ in the limit $d\to 0$. A global heat bath is still described by Eq. (2)with $f^{(g)}_{kq}=1$.

From these rates we computed the corresponding mean-occupations for a system of $N=0.1L/d$ particles, length $L=500d$, and needle position $\ell=20d$. In Fig. 3 we plot the condensate fraction, i.e. the fraction of particles occupying the most occupied mode whose wave number is indicated by the color of the shading (green for $k_{0}\approx 0$ and light blue for $k_{1}\approx\pi d/\ell$). As for the tight-binding chain, we can clearly see that the temperature of the global bath at which the system condenses increases with needle temperature. As a result Bose condensation is found for a system coupled to two baths both having temperatures well above the equilibrium condensation temperature $T_{c}^{\text{ne}}$ (which is indicated as blue-white dotted line). Moreover, when the needle temperature is increased further, ground-state condensation in mode $k_{0}\approx 0$ is superseded by the formation of a condensate in the excited mode $k_{1}\approx\pi d/\ell$, which provides a better decoupling from the hot needle.