Coherent macroscopic quantum tunneling in boson-fermion mixtures

TL;DR

This paper predicts that in cold atom systems with Bose-Einstein condensates and fermionic atoms, the BEC can coherently tunnel between separated and mixed states, offering a new platform to study macroscopic quantum tunneling.

Contribution

Theoretical demonstration of coherent macroscopic quantum tunneling in BEC-fermion mixtures, with specific predictions for experimental parameters and observable oscillations.

Findings

Coherent oscillations of BEC in fermionic environment predicted

Period, amplitude, and dissipation rate calculated for specific isotopes

Potential for experimental observation discussed

Abstract

We show that the cold atom systems of simultaneously trapped Bose-Einstein condensates (BEC's) and quantum degenerate fermionic atoms provide promising laboratories for the study of macroscopic quantum tunneling. Our theoretical studies reveal that the spatial extent of a small trapped BEC immersed in a Fermi sea can tunnel and coherently oscillate between the values of the separated and mixed configurations (the phases of the phase separation transition of BEC-fermion systems). We evaluate the period, amplitude and dissipation rate for $^{23}$Na and $^{40}$K-atoms and we discuss the experimental prospects for observing this phenomenon.

Figures

Equations

$$S=S_{BEC}+\mathrm{Tr}\log \left[\hslash {\partial }_{\tau }-\frac{{\hslash }^2{\nabla }^2}{2m_F}-{\mu }_F+{\lambda }_{BF}|{\Psi }_B{|}^2\right],$$

$$\frac{\delta \mathrm{Tr}\log [...]}{\delta {\Psi }_B^{\ast }}=\frac{{\lambda }_{BF}{k}_F^3}{3{\pi }^2}\mathrm{Re}{\left[1-\frac{{\lambda }_{BF}|{\Psi }_B(\mathbf{r}){|}^2}{{\mu }_F}\right]}^{3/2}{\Psi }_B,$$

$$\displaystyle H=\int d^{3}{\bf r}\Big{[}{\hbar^{2}\over 2m_{B}}|\nabla\Psi_{B}|^{2}+{m_{B}\omega_{T}^{2}{\bf r}^{2}\over 2}|\Psi_{B}|^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$$

$$\displaystyle+{1\over 2}(\lambda_{BB}-\lambda^{\prime})|\Psi_{B}|^{4}+{\eta\over 6}|\Psi_{B}|^{6}\Big{]},$$

$${\Psi }_B^0(\mathbf{r})=\frac{N_B^{\frac{1}{2}}}{{\pi }^{\frac{3}{4}}(x{R}_0{)}^{\frac{3}{2}}}\exp \left[-\frac{{\mathbf{r}}^2}{2(x{R}_0{)}^2}\right],$$

$$E_0(x)=\frac{3N_B\hslash {\omega }_T}{2}\left(\frac{1}{x^2}+\frac{x^2}{4}-\frac{\alpha }{3x^3}+\frac{\beta }{6x^6}\right),$$

$$S_0[x(\tau )]=\int d\tau \left[\frac{m_0{\dot{x}}^2}{2}+E_0(x)\right],$$

$$\left[-\frac{{\partial }^2}{\partial x^2}+\frac{x^2}{4}-a\frac{{\varphi }^2}{x^2}+\frac{b}{4}\frac{{\varphi }^4}{x^4}\right]\varphi =\mu \varphi ,$$

$$-\frac{{\lambda }_{BF}^2}{2\hslash }\int \frac{d\omega }{2\pi }\int \frac{d^3\mathbf{q}}{(2\pi {)}^3}{\chi }_0(\mathbf{q},\omega )|{\rho }_B(\omega ,\mathbf{q}){|}^2.$$

$$S_{\mathrm{diss}}=\gamma \hslash \int d\tau d{\tau }^{\prime }{\sigma }_z(\tau ){\sigma }_z({\tau }^{\prime })(\tau -{\tau }^{\prime }{)}^{-2},$$

Full text

The tunneling of a macroscopic (or collective) variable of a many-body system through a classically forbidden region, macroscopic quantum tunneling (MQT), is a phenomenon of fundamental interest Leggett and a recurring theme in a variety of fields ranging from nuclear (fission) and condensed matter physics (e.g. quantum magnets qm , SQUIDs) to quantum optics (macroscopic Schrodinger cat states qo and beyond-standard limit measurements). Nevertheless, stringent tests under well-understood and controlled conditions remain an experimental challenge. Cold atom gases, arguably the cleanest and best understood mesoscopic systems which, furthermore, offer unprecedented control knobs such as the ability to vary the inter-particle interactions Fesh , now provide an intriguing candidate laboratory for the study of MQT.

The first cold atom MQT proposals theory1 suggested observing the collapse of a trapped dilute gas Bose-Einstein condensate (BEC) of mutually attracting bosons. However, the experimental results Hulet ; Wieman were either too sensitive to particle number to distinguish MQT from classical collapse Hulet , or the analysis was complicated by more complex dynamics (such as ’clumping’) Wieman . Evidence of coherence (of the many-body system taking on a linear superposition of states that correspond to the macroscopic variable residing on either side of the barrier) is even more difficult to gather. Such coherence would be more readily observable in the MQT between ${\it long}$-${\it lived}$ states, in which case one could set up a coherent population oscillation between the many-body states. Such long-lived states naturally occur in (zero-temperature) first order phase transitions in which the order parameter, which provides the macroscopic variable, can tunnel through the barrier of its Landau-Ginzburg potential. In the infinite system limit, the coupling between the two states rigorously vanishes, but finite- size cold atom systems of moderate particle numbers provide, once again, a promising candidate to observe the MQT coherence between states of different phases, as we show below.

An earlier proposal to observe MQT between states in which the components of a BEC-mixture arrange themselves differently in space, involved a very low coupling on account of the small spatial overlap between the single component densities in the different states Kasamatsu . In this paper, we propose that MQT can be realized and its coherence, perhaps, observed in trapped gas mixtures of a single-component fermion system and a BEC. Such mixtures are currently created sympath e.g. in the sympathetic cooling scheme in which the colder BEC cools the fermions. The tunneling and coherent oscillations that we target would occur between states of the mixed and separated phases in the phase separation transition of the fermion-BEC mixture Molmer . Such transitions could be accessed by varying the scattering length of the boson-fermion interaction Simoni .

We consider $N_{B}$ atomic bosons confined in a spherically symmetric harmonic trap (of frequency $\omega_{T}$) interacting with a much larger system of atomic fermions. For simplicity we assume the fermions to occupy an infinite volume. The Hamiltonian of the bosons is described by the standard Gross-Pitaevskii (GP) form Leggett , i.e., with inter-particle interactions described by a contact potential ($\propto\lambda_{BB}\delta({\bf r}-{\bf r}^{\prime})$), which we choose to be repulsive ($\lambda_{BB}>0$) We assume that the interaction of bosons with fermions is also contact-like, contributing $\lambda_{BF}|\Psi_{B}|^{2}|\Psi_{F}|^{2}$ to the Hamiltonian density, where $\lambda_{BF}$ is the fermion-boson coupling constant. Furthermore, all fermions occupy in the same spin state so that the short-range inter-fermion interactions do not contribute by virtue of the Pauli exclusion principle.

We are interested in the dynamics of the reduced system of bosons described by the functional

[TABLE]

where $S_{0}$ is the action of the bosons alone, $S_{BEC}=\int d\tau(\hbar\int d^{3}{\bf r}{\dot{\Psi}}_{B}\Psi_{B}^{\ast}-H_{BEC})$, and the second term is a contribution due to the interaction of bosons with fermions; $\mu_{F}$ is the chemical potential of the fermions. Here and throughout the paper we will be utilizing the imaginary time (Matsubara) representation, unless stated otherwise. An explicit evaluation of the second term is a challenging task. However, here we are interested in the dynamics of the slow breathing mode of the BEC $\Psi_{B}^{0}$, which can be treated in the self-similar density approximation. This dynamics describes the longitudinal expansions (and contractions) of the condensate. Finite size effects such as the appearance of a non-vanishing excitation energy (gap) can decouple this mode from other excitation modes. Hence, $\Psi_{B}^{0}$ peaks at small frequencies ($\omega$) and small wavevectors (${\bf q}$), giving a $\Psi_{B}^{0}$ that is a slowly varying function of spacial and temporal coordinates. In such case the ${\rm Tr}\log[...]$ in Eq. (1) can be evaluated within the Thomas-Fermi approximation. A straightforward zero-temperature calculation yields

[TABLE]

where $k_{F}$ is the Fermi wavevector. Eq. (2) represents an additional term in the Gross-Pitaevskii (GP) equation, $\delta S_{GP}/\delta\Psi_{B}^{\ast}=0$, resulting from interaction with fermions. In order to analyze the physical meaning of Eq. (2) let us expand it in powers of $\Psi_{B}$. The first nontrivial contribution is a term $-2\lambda^{\prime}|\Psi_{B}|^{2}\Psi_{B}$, $\lambda^{\prime}=(\lambda_{BF}^{2}k_{F}^{3}/4\pi^{2}\mu_{F})$, which corresponds to the attraction between bosons mediated by interaction with fermions. For nonzero, but small $\omega$ and $\bf q$ there is an additional term (of the order of $\lambda_{BF}^{2}$) related to the dissipation of the condensate due to the Landau damping, as we discuss below. The next order yields $\eta|\Psi_{B}|^{4}\Psi_{B}$, $\eta=(k_{F}^{3}\lambda_{BF}^{3}/8\pi^{2}\mu_{F}^{2})$. Unlike the previous term this one is positive, and represents reduction in the effective boson-boson attraction due to depletion of fermions in the regions of high density of the bosons. The next order terms (in $\lambda_{BF}$) prove to be unimportant as can be verified directly from Eq. (2). Therefore we will replace the potential energy contribution in GP equation given by Eq. (2) by the two terms discussed above com .

To analyze the dynamics of the slow (breathing) mode described by the Hamiltonian

[TABLE]

we apply the time dependent variational principle. Since we are interested in ground state properties of Eq. (5) we use a spherically symmetric Gaussian trial wavefunction

[TABLE]

parameterized by a dimensionless parameter $x$ that characterizes the BEC’s spatial width in units of the zero point motion amplitude $R_{0}=(\hbar/2m_{B}\omega_{T})^{1/2}$. Substitution of this wavefunction into Eq. (5) yields the following dependence of the ground-state energy $E^{0}$ on $x$:

[TABLE]

where $\alpha=N_{B}(\lambda^{\prime}-\lambda_{BB})/[(2\pi)^{3/2}R_{0}^{3}\hbar\omega_{T}]$ and $\beta=4N_{B}^{2}\eta/(3^{5/2}\pi^{3}R_{0}^{6}\hbar\omega_{T})$.

For positive but relatively small $\alpha$, i.e., for $\alpha<\alpha_{cr}=32(2/5)^{1/4}/15\simeq 1.69$, $E_{0}$ may develop two competing minima, depending on the value of the $\beta$-parameter. The energy barrier separating the minima is caused by the same effect as the barrier appearing in the description of a BEC with attractive interactions: it arises due to the competition between the kinetic and the interaction energies, i.e, the first and the third terms in the right-hand side (rhs) of Eq. (5). In the absence of the last term in the rhs of Eq. (5) the state in this well would have been metastable - the energy would tend to $-\infty$ at $x\rightarrow 0$. The $1/x^{6}$ term stabilizes the system: for small $x$ this term rapidly increases, giving rise to another minimum of $E_{0}(x)$, now due to the competition between the last two terms in the rhs of Eq. (5).

At certain values of $\alpha$ and $\beta$ the two minima of $E_{0}(x)$ will have the same energy, and the ground state of the system becomes degenerate. Since our system is finite, this degeneracy will be lifted by the quantum tunneling transition between the two states. Such mechanism has been suggested to be the dominant decay process for condensates with attractive interactions between particles theory1 . The tunneling corresponds to the low energy excitations of the breathing mode, described by the wavefunction in Eq. (4). It has been shown in theory1 that by accounting for the superfluid motion of the condensate (which can be done by introducing a phase-factor $e^{i\phi}$ for the wavefunction in Eq. (4) and requiring the superfluid velocity ${\bf v}_{s}=(\hbar/m_{B}){\bf\nabla}\phi$ to satisfy the continuity equation) one obtains an effective action for the breathing mode of the condensate

[TABLE]

where $E_{0}(x)$ is given by Eq. (5) and $m_{0}=3m_{B}N_{B}R_{0}^{2}/2$. Thus the dynamics of the ground state wavefunction of the condensate is that of a quantum particle of mass $m_{0}$ moving in the potential $E_{0}(x)$.

A direct analysis of the Shrodinger equation corresponding to Eq. (6), however, is quite cumbersome since the two wells are generally quite asymmetric. Instead we choose an alterative route: we compute the ground state energy and obtain the tunneling rate by numerically solving the time-independent GP equation, $\delta H/\delta\Psi_{B}=E\Psi_{B}$, where $H$ is given by Eq. (3). The latter approach also serves as an independent justification of the variational method and confirms that macroscopic quantum tunneling, QMT, is the mechanism that causes the transition between the two states of the condensate. Upon substitution $\Psi_{B}\sim\phi/r$, the time-independent GP equation can be cast in the form

[TABLE]

where the $\phi(x)$-function is normalized to unity, $a=(\pi/2)^{1/2}\alpha$, $b=3^{5/2}\pi\beta/16$, and $x=r/R_{0}$, $\mu=E/\hbar\omega_{T}$. We find the ground state numerically by replacing the rhs of Eq. (7) by $-\partial_{\tau}\phi$ and propagating $\phi$ in the imaginary time $\tau$ until it converges to the ground state $\phi_{0}$ (or $\Psi_{B}^{0}$). We then evaluate the ground state energy according to Eq. (3) and present the results in Fig. 2(a) as a function of the $b$-parameter for different values of $a$. Fig. 2(b) shows the dispersion of the ground state width, $(1/N_{B})\int d^{3}{\bf r}|\Psi_{B}^{0}({\bf r})|^{2}{\bf r}^{2}$, as a function of those same parameters. For $a<a_{cr}=1.83$ the ground state energy and dispersion undergo a sharp crossover between the state with compressed and expanded BEC wavefunctions (corresponding to the phase separated and mixed states) as functions of $b$. Note that the value $a_{cr}$ corresponds to the value of $\alpha_{cr}=1.46$, which is quite close to the above critical value of $1.69$ obtained from the variational approach. The dependence of ground state energy near the critical value of $a$ is shown in the inset of Fig. 2(a). Clearly the ground state energy exhibits avoided level crossing, which is in accordance with the above conjecture (e.g., Eq. (6)) of macroscopic quantum tunneling between the two local energy minima.

The value of the tunneling matrix element $\Delta$ between two local “ground” states $\epsilon_{1}$ and $\epsilon_{2}$ can be deduced by fitting the calculated energy curves in Fig. 1 with the standard expression Landau , $\epsilon=(\epsilon_{1}+\epsilon_{2})/2-[(\epsilon_{1}-\epsilon_{2})^{2}/4-\Delta^{2}]^{1/2}$ and assuming that in the vicinity of the point of crossover both $\epsilon_{1}$ and $\epsilon_{2}$ are linear functions of parameter $b$. For $a=1.81$ one finds $\Delta\sim 10^{-4}\times\hbar\omega_{T}$, while for $a=1.82$, $\Delta\sim 10^{-2}\times\hbar\omega_{T}$. Assuming that the ground state wavefunctions have Gaussian shape, $|\Psi_{B}|^{2}_{1(2)}\sim R^{-3}_{1(2)}\exp(-{\bf r}^{2}/R^{2}_{1(2)})$, from Fig. 2 one finds that ${\bar{R}}=(R_{1}+R_{2})/2\simeq 0.85R_{0}$ for both $a=1.81$ and $a=1.82$, and $\delta R=|R_{1}-R_{2}|\simeq 0.09R_{0}$ for $a=1.81$ and $\delta R\simeq 0.03R_{0}$ for $a=1.82$. For a typical value of the trapping frequency $\nu_{T}=10^{2}Hz$ ($\omega_{T}=2\pi\nu_{T}$), the two tunneling rates are $\Delta_{1.81}/\hbar=10^{-2}\,s^{-1}$ and $\Delta_{1.82}/\hbar=10^{2}\,s^{-1}$. Since the value of $R_{0}$ for most trapped atomic BEC’s is of the order of a few microns, the difference between the radii of the two condensate states $\delta R$ is submicron. Such small variation may be difficult to observe in situ by optical means. However, the expansion process that takes place in time-of-flight measurements after the trap potential is shut off and the expanding atoms are observed, has successfully magnified small distance features in other experiments.

Role of dissipation: The above analysis determines the tunneling rate, but does not address the question whether the tunneling process is quantum coherent. Will the probability of the system to occupy one of the two macroscopic states oscillate in time as $\cos^{2}{(\Delta\,t/\hbar)}$? The fermions not only provide the BEC with the effective interaction, they also cause fluctuations which can destroy the macroscopic quantum coherence. To evaluate the effect of fluctuations, it is sufficient to consider the first non-vanishing frequency-dependent contribution into the effective action of the bosons coming from the perturbative expansion of the ${\rm Tr\log{[...]}}$ term in Eq. (1):

[TABLE]

Here $\rho_{B}({\bf q},\omega)$ is the Fourier transform of $\rho_{B}({\bf r},t)$ and $\chi_{0}$ is the response function of the non-interacting fermions. In the small frequency domain $\chi_{0}=(1/4\pi)[\hbar^{2}k_{F}^{3}/(\pi\mu_{F})+m_{F}^{2}|\omega|/(\hbar^{2}q)]$. The frequency-independent part of $\chi_{0}$ has already been incorporated in the effective interaction between bosons, i.e., $\lambda^{\prime}|\Psi_{B}|^{4}$ term in Eq. (3). The second term in $\chi_{0}$ is responsible for damping. To quantify its role we employ a two-state approximation in describing the tunneling dynamics. In this representation the tunneling is described by the Hamiltonian $H_{\rm tun}=\Delta{\hat{\sigma}}_{x}$, where ${\hat{\sigma}}_{x}$ is a Pauli matrix with non-zero off-diagonal elements, and the position operator, i.e. the spatial width of the ground-state BEC wavefunction, is given by ${\hat{R}}={\bar{R}}+(\delta R/2){\hat{\sigma}}_{z}$, where ${\hat{\sigma}}_{z}$ is the diagonal Pauli matrix (with $\pm 1$ along the diagonal). The dissipative part of the action for $H_{\rm tun}$ can be derived from Eq. (8) by substituting a Gaussian ansatz, $\rho_{B}({\bf r},t)=N_{B}/[\pi^{3/2}R^{3}(t)]\exp{[-{\bf r}^{2}/R^{2}(t)]}$, where $R(t)={\bar{R}}+(\delta R/2)\sigma_{z}(t)$, $\sigma_{z}=\pm 1$, into Eq. (8). For $\delta R\ll{\bar{R}}$ one obtains

[TABLE]

where $\gamma=N_{B}^{2}\lambda_{BF}^{2}m_{F}^{2}\delta R^{2}/[2(2\pi\hbar)^{4}{\bar{R}}^{4}]$. Eq. (9), together with $H_{\rm tun}$ defined above, describes dissipative dynamics of a two-state system. Such dynamics has been extensively studied in connection with macroscopic quantum tunneling of a superconducting phase in Josephson junctions, and is known to depend critically on the value the parameter $\gamma$. Specifically, for $\gamma>1$ the two-state oscillation is always overdamped and at zero temperature it exhibits localization as a result of quantum fluctuations leggett . It is therefore instructive to evaluate $\gamma$ for our situation. For estimates we consider an atomic mixture of 23Na (bosons) and 40K (fermions), which have natural scattering lengthes $a_{BB}\simeq 1\,nm$ ($\lambda_{BB}=4\pi\hbar^{2}a_{BB}/m_{B}$) and $a_{BF}\simeq 4\,nm$ ($\lambda_{BF}=2\pi\hbar^{2}a_{BF}[(1/m_{B})+(1/m_{F})]$). For these data we obtain a critical value of $N_{B}^{cr}\simeq 12400$ (again for $\nu_{T}=10^{2}Hz$) and the fermion density $n_{F}^{cr}\simeq 7.4\times 10^{15}cm^{-3}$. Then, for $a=1.81$ we obtain $\gamma_{1.81}\simeq 1.1$, which corresponds to the localized case (at $T=0$), whereas for $a=1.82$ one gets $\gamma_{1.82}\simeq 0.1$. In the high temperature limit (for $k_{B}T>\Delta$) the relaxation rate $\Gamma$ can be expressed in terms of $\gamma$ as $\hbar\Gamma=\pi\gamma k_{B}T$ leggett , and therefore coherent (underdamped) oscillations can be observed for $T\ll\Delta_{1.82}/(\gamma_{1.82}k_{B})=0.5nK$. The situation can be improved, however, if one utilizes a Feshbach resonance Fesh to increase the $a_{BF}$ scattering length. For example, for $a_{BF}=80nm$ one finds $N_{B}^{cr}\simeq 25$ and $n_{F}^{cr}\simeq 2.6\times 10^{11}cm^{-3}$, and $\gamma_{1.82}\simeq 2.5\times 10^{-4}$. For such parameters coherent oscillations can be observed for $T\ll 0.2\mu K$, which is easily observable. A low particle number also reduces the uncertainty of an atomic counting measurement that can be carried out in the time-of-flight procedure 25 .

In summary we argue that a trapped boson-fermion mixture can exhibit MQT tunneling and coherent oscillations. Our studies indicate that MQT can be observed in 23Na and 40K atomic mixtures of sufficiently low temperatures.

We thank M. Boshier and S. A. Gurvitz for valuable discussions. The work is supported by the US DOE.