Coherent macroscopic quantum tunneling in boson-fermion mixtures
D. Mozyrsky, I. Martin, and E. Timmermans
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545
(Dated: August 11, 2021)
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 23Na and 40K-atoms and we discuss the
experimental prospects for observing this phenomenon.
PACS numbers: 05.30.Jp, 03.75.Kk, 32.80.Pj, 67.90.+z
The tunneling of a macroscopic (or collective) vari-
able of a many-body system through a classically forbid-
den region, macroscopic quantum tunneling (MQT), is a
phenomenon of fundamental interest [1] and a recurring
theme in a variety of fields ranging from nuclear (fission)
and condensed matter physics (e.g. quantummagnets [2],
SQUIDs) to quantum optics (macroscopic Schrodinger
cat states [3] 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 under-
stood mesoscopic systems which, furthermore, offer un-
precedented control knobs such as the ability to vary the
inter-particle interactions [4], now provide an intriguing
candidate laboratory for the study of MQT.
The first cold atom MQT proposals [5] suggested ob-
serving the collapse of a trapped dilute gas Bose-Einstein
condensate (BEC) of mutually attracting bosons. How-
ever, the experimental results [6, 7] were either too sensi-
tive to particle number to distinguish MQT from classi-
cal collapse [6], or the analysis was complicated by more
complex dynamics (such as ’clumping’) [7]. Evidence of
coherence (of the many-body system taking on a linear
superposition of states that correspond to the macro-
scopic 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 long-lived
states, in which case one could set up a coherent pop-
ulation oscillation between the many-body states. Such
long-lived states naturally occur in (zero-temperature)
first order phase transitions in which the order parame-
ter, 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 sys-
tems of moderate particle numbers provide, once again,
a promising candidate to observe the MQT coherence be-
tween 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 them-
selves differently in space, involved a very low coupling
on account of the small spatial overlap between the sin-
gle component densities in the different states [8]. 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 [9] e.g. in the sympa-
thetic 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 [10]. Such transitions could be
accessed by varying the scattering length of the boson-
fermion interaction [11].
We consider NB atomic bosons confined in a spheri-
cally symmetric harmonic trap (of frequency ωT ) inter-
acting with a much larger system of atomic fermions.
For simplicity we assume the fermions to occupy an in-
finite volume. The Hamiltonian of the bosons is de-
scribed by the standard Gross-Pitaevskii (GP) form [1],
i.e., with inter-particle interactions described by a con-
tact potential (∝ λBBδ(r − r
′)), which we choose to be
repulsive (λBB > 0) We assume that the interaction of
bosons with fermions is also contact-like, contributing
λBF |ΨB|
2|ΨF |
2 to the Hamiltonian density, where λBF
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 sys-
tem of bosons described by the functional
S = SBEC +Tr log
~∂τ −
− µF + λBF |ΨB|
where S0 is the action of the bosons alone, SBEC =
d3rΨ̇BΨ
B − HBEC), and the second term is
a contribution due to the interaction of bosons with
fermions; µF 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 inter-
ested in the dynamics of the slow breathing mode of the
BEC Ψ0B, which can be treated in the self-similar density
http://arxiv.org/abs/0704.0650v1
approximation. This dynamics describes the longitudinal
expansions (and contractions) of the condensate. Finite
size effects such as the appearance of a non-vanishing ex-
citation energy (gap) can decouple this mode from other
excitation modes. Hence, Ψ0B peaks at small frequen-
cies (ω) and small wavevectors (q), giving a Ψ0B that is
a slowly varying function of spacial and temporal coor-
dinates. In such case the Tr log[...] in Eq. (1) can be
evaluated within the Thomas-Fermi approximation. A
straightforward zero-temperature calculation yields
δTr log[...]
λBF |ΨB(r)|
ΨB, (2)
where kF is the Fermi wavevector. Eq. (2) represents an
additional term in the Gross-Pitaevskii (GP) equation,
δSGP /δΨ
B = 0, resulting from interaction with fermions.
In order to analyze the physical meaning of Eq. (2) let
us expand it in powers of ΨB. The first nontrivial con-
tribution is a term −2λ′|ΨB|
2ΨB, λ
′ = (λ2BF k
F /4π
2µF ),
which corresponds to the attraction between bosons me-
diated by interaction with fermions. For nonzero, but
small ω and q there is an additional term (of the or-
der of λ2BF ) related to the dissipation of the condensate
due to the Landau damping, as we discuss below. The
next order yields η|ΨB|
4ΨB, η = (k
BF /8π
2µ2F ). Un-
like 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 λBF ) prove to
be unimportant as can be verified directly from Eq. (2).
Therefore we will replace the potential energy contribu-
tion in GP equation given by Eq. (2) by the two terms
discussed above [12].
To analyze the dynamics of the slow (breathing) mode
described by the Hamiltonian
|∇ΨB|
2 (3)
(λBB − λ
′)|ΨB|
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
Ψ0B(r) =
4 (xR0)
2(xR0)2
, (4)
parameterized by a dimensionless parameter x that char-
acterizes the BEC’s spatial width in units of the zero
point motion amplitude R0 = (~/2mBωT )
1/2. Substitu-
tion of this wavefunction into Eq. (5) yields the following
dependence of the ground-state energy E0 on x:
E0(x) =
3NB~ωT
, (5)
where α = NB(λ
′ − λBB)/[(2π)
3/2R30~ωT ] and β =
4N2Bη/(3
5/2π3R60~ωT ). For positive but relatively small
FIG. 1: Density profiles of the bosons (solid lines 1 and 2) and
corresponding density profiles of the fermions (dashed lines
1’ and 2’) in two metastable states: (1) with fermions having
zero density at the center of the trap (“separated phase”) and
(2) with nonzero density of fermions (“mixed phase”). The
dotted line represents schematically an effective potential for
the breathing mode of the bosons.
α, i.e., for α < αcr = 32(2/5)
1/4/15 ≃ 1.69, E0 may de-
velop two competing minima, depending on the value of
the β-parameter. The energy barrier separating the min-
ima 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
−∞ at x → 0. The 1/x6 term stabilizes the system: for
small x this term rapidly increases, giving rise to another
minimum of E0(x), now due to the competition between
the last two terms in the rhs of Eq. (5).
At certain values of α and β the two minima of E0(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 tunnel-
ing transition between the two states. Such mechanism
has been suggested to be the dominant decay process for
condensates with attractive interactions between parti-
cles [5]. The tunneling corresponds to the low energy ex-
citations of the breathing mode, described by the wave-
function in Eq. (4). It has been shown in [5] that by
accounting for the superfluid motion of the condensate
(which can be done by introducing a phase-factor eiφ for
the wavefunction in Eq. (4) and requiring the superfluid
velocity vs = (~/mB)∇φ to satisfy the continuity equa-
tion) one obtains an effective action for the breathing
mode of the condensate
S0[x(τ)] =
+ E0(x)
, (6)
where E0(x) is given by Eq. (5) and m0 = 3mBNBR
Thus the dynamics of the ground state wavefunction of
the condensate is that of a quantum particle of mass m0
moving in the potential E0(x).
A direct analysis of the Shrodinger equation corre-
sponding 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, δH/δΨB =
EΨB, where H is given by Eq. (3). The latter approach
also serves as an independent justification of the varia-
tional method and confirms that macroscopic quantum
tunneling, QMT, is the mechanism that causes the tran-
sition between the two states of the condensate. Upon
substitution ΨB ∼ φ/r, the time-independent GP equa-
tion can be cast in the form
φ = µφ, (7)
where the φ(x)-function is normalized to unity, a =
(π/2)1/2α, b = 35/2πβ/16, and x = r/R0, µ = E/~ωT .
We find the ground state numerically by replacing the
rhs of Eq. (7) by −∂τφ and propagating φ in the imag-
inary time τ until it converges to the ground state φ0
(or Ψ0B). We then evaluate the ground state energy ac-
cording 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/NB)
d3r|Ψ0B(r)|
2, as a function of those same pa-
rameters. For a < acr = 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 acr corresponds
to the value of αcr = 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 conjec-
ture (e.g., Eq. (6)) of macroscopic quantum tunneling
between the two local energy minima.
The value of the tunneling matrix element ∆ between
two local “ground” states ǫ1 and ǫ2 can be deduced by
fitting the calculated energy curves in Fig. 1 with the
standard expression [13], ǫ = (ǫ1+ ǫ2)/2− [(ǫ1− ǫ2)
∆2]1/2 and assuming that in the vicinity of the point
of crossover both ǫ1 and ǫ2 are linear functions of pa-
rameter b. For a = 1.81 one finds ∆ ∼ 10−4 × ~ωT ,
while for a = 1.82, ∆ ∼ 10−2 × ~ωT . Assuming that
the ground state wavefunctions have Gaussian shape,
∼ R−3
exp(−r2/R2
), from Fig. 2 one finds
that R̄ = (R1 + R2)/2 ≃ 0.85R0 for both a = 1.81 and
a = 1.82, and δR = |R1 − R2| ≃ 0.09R0 for a = 1.81
and δR ≃ 0.03R0 for a = 1.82. For a typical value
of the trapping frequency νT = 10
2Hz (ωT = 2πνT ),
the two tunneling rates are ∆1.81/~ = 10
−2 s−1 and
∆1.82/~ = 10
2 s−1. Since the value of R0 for most
trapped atomic BEC’s is of the order of a few microns,
FIG. 2: (a) Dependence of the ground state energy of the BEC
(per particle, in units of ~ωT ) as a function of parameters a
and b; (b) Dispersion of the ground state spatial extent as a
function of the same parameters.
the difference between the radii of the two condensate
states δR 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 cos2 (∆ t/~)? The
fermions not only provide the BEC with the effective in-
teraction, 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 perturba-
tive expansion of the Tr log [...] term in Eq. (1):
(2π)3
χ0(q, ω)|ρB(ω,q)|
2. (8)
Here ρB(q, ω) is the Fourier transform of ρB(r, t) and
χ0 is the response function of the non-interacting
fermions. In the small frequency domain χ0 =
(1/4π)[~2k3F /(πµF ) + m
F |ω|/(~
2q)]. The frequency-
independent part of χ0 has already been incorporated
in the effective interaction between bosons, i.e., λ′|ΨB|
term in Eq. (3). The second term in χ0 is responsible
for damping. To quantify its role we employ a two-
state approximation in describing the tunneling dynam-
ics. In this representation the tunneling is described by
the Hamiltonian Htun = ∆σ̂x, where σ̂x is a Pauli ma-
trix with non-zero off-diagonal elements, and the posi-
tion operator, i.e. the spatial width of the ground-state
BEC wavefunction, is given by R̂ = R̄+(δR/2)σ̂z, where
σ̂z is the diagonal Pauli matrix (with ±1 along the di-
agonal). The dissipative part of the action for Htun
can be derived from Eq. (8) by substituting a Gaus-
sian ansatz, ρB(r, t) = NB/[π
3/2R3(t)] exp [−r2/R2(t)],
where R(t) = R̄ + (δR/2)σz(t), σz = ±1, into Eq. (8).
For δR ≪ R̄ one obtains
Sdiss = γ~
dτdτ ′σz(τ)σz(τ
′)(τ − τ ′)−2, (9)
where γ = N2Bλ
2/[2(2π~)4R̄4]. Eq. (9), to-
gether with Htun defined above, describes dissipative dy-
namics of a two-state system. Such dynamics has been
extensively studied in connection with macroscopic quan-
tum tunneling of a superconducting phase in Josephson
junctions, and is known to depend critically on the value
the parameter γ. Specifically, for γ > 1 the two-state os-
cillation is always overdamped and at zero temperature
it exhibits localization as a result of quantum fluctua-
tions [14]. It is therefore instructive to evaluate γ for our
situation. For estimates we consider an atomic mixture
of 23Na (bosons) and 40K (fermions), which have natural
scattering lengthes aBB ≃ 1nm (λBB = 4π~
2aBB/mB)
and aBF ≃ 4nm (λBF = 2π~
2aBF [(1/mB) + (1/mF )]).
For these data we obtain a critical value of N crB ≃ 12400
(again for νT = 10
2Hz) and the fermion density ncrF ≃
7.4×1015cm−3. Then, for a = 1.81 we obtain γ1.81 ≃ 1.1,
which corresponds to the localized case (at T = 0),
whereas for a = 1.82 one gets γ1.82 ≃ 0.1. In the high
temperature limit (for kBT > ∆) the relaxation rate Γ
can be expressed in terms of γ as ~Γ = πγkBT [14],
and therefore coherent (underdamped) oscillations can
be observed for T ≪ ∆1.82/(γ1.82kB) = 0.5nK. The
situation can be improved, however, if one utilizes a Fes-
hbach resonance [4] to increase the aBF scattering length.
For example, for aBF = 80nm one finds N
B ≃ 25 and
ncrF ≃ 2.6 × 10
11cm−3, and γ1.82 ≃ 2.5 × 10
−4. For
such parameters coherent oscillations can be observed for
T ≪ 0.2µK, which is easily observable. A low particle
number also reduces the uncertainty of an atomic count-
ing measurement that can be carried out in the time-of-
flight procedure [15].
In summary we argue that a trapped boson-fermion
mixture can exhibit MQT tunneling and coherent oscil-
lations. Our studies indicate that MQT can be observed
in 23Na and 40K atomic mixtures of sufficiently low tem-
peratures.
We thank M. Boshier and S. A. Gurvitz for valuable
discussions. The work is supported by the US DOE.
[1] A. J. Leggett et al., Rev. Mod. Phys., 59, 1 (1998)
[2] L. Gunther and B. Barbera, Eds. ’Quantum tunneling
of magnetization - QTM’94 (Kluwer, Dordrecht, Nethre-
lands, 1995).
[3] J. I. Cirac et al., Phys. Rev A, 57, 1208 (1998).
[4] E. Timmermans et al, Phys. Rep., 315, 199 (1999).
[5] H. T. C. Stoof, J. Stat. Phys. 87, 1353 (1997); M. Ueda
and A. J. Leggett, Phys. Rev. Lett., 80, 1576 (1998); J.
A. Freire and D. P. Arovas, Phys. Rev. A 59, 1461 (1999);
C. Huepe, S. Metes, G. Dewel, P. Borckmans, and M. E.
Brachet, Phys. Rev. Lett., 82, 1616 (1999).
[6] C. A. Sackett et al, Phys. Rev. Lett. 82, 876 (1999); J.
M. Gerton et al, Nature;408, 692 (2000).
[7] E. A. Donley et al., Nature; 412, 295 (2001); J. L.
Roberts et al. Phys. Rev. Lett.; 86, 4211 (2001).
[8] K. Kasamatsu et al., Phys. Rev. A, 64, 053605 (2001).
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Schreck et al., Phys. Rev. Lett. 87 080403 (2001); G.
Modugno et al., Science, 297 2240 (2002); M. W. Zwier-
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Phys. Rev. Lett. 93, 143001 (2004).
[10] The spatial arrangements in the fermion-BEC mixtures
were first discussed in K. Molmer, Phys. Rev. Lett., 80,
1804 (1998), and the infinite system phase separation
transition was described in L. Viverit et al. Phys. Rev.
A, 61, 053605 (2000).
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tial given by Eq. (2) for will be presented elsewhere.
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Pergamon Press (1965).
[14] A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987).
[15] Low particle numbers can be measured very accurately
with resonance fluorescence, for instance, see D. Frese
et al., Phys. Rev. Lett., 85, 3777 (2000); recent work
also illustrated subPoissonian counting (∆N <
N) for
larger numbers, T. Campey et al., Phys. Rev. A, 74,
043612 (2006).

  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.

Coherent macroscopic quantum tunneling in boson-fermion mixtures
D. Mozyrsky, I. Martin, and E. Timmermans
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545
(Dated: August 11, 2021)
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 23Na and 40K-atoms and we discuss the
experimental prospects for observing this phenomenon.
PACS numbers: 05.30.Jp, 03.75.Kk, 32.80.Pj, 67.90.+z
The tunneling of a macroscopic (or collective) vari-
able of a many-body system through a classically forbid-
den region, macroscopic quantum tunneling (MQT), is a
phenomenon of fundamental interest [1] and a recurring
theme in a variety of fields ranging from nuclear (fission)
and condensed matter physics (e.g. quantummagnets [2],
SQUIDs) to quantum optics (macroscopic Schrodinger
cat states [3] 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 under-
stood mesoscopic systems which, furthermore, offer un-
precedented control knobs such as the ability to vary the
inter-particle interactions [4], now provide an intriguing
candidate laboratory for the study of MQT.
The first cold atom MQT proposals [5] suggested ob-
serving the collapse of a trapped dilute gas Bose-Einstein
condensate (BEC) of mutually attracting bosons. How-
ever, the experimental results [6, 7] were either too sensi-
tive to particle number to distinguish MQT from classi-
cal collapse [6], or the analysis was complicated by more
complex dynamics (such as ’clumping’) [7]. Evidence of
coherence (of the many-body system taking on a linear
superposition of states that correspond to the macro-
scopic 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 long-lived
states, in which case one could set up a coherent pop-
ulation oscillation between the many-body states. Such
long-lived states naturally occur in (zero-temperature)
first order phase transitions in which the order parame-
ter, 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 sys-
tems of moderate particle numbers provide, once again,
a promising candidate to observe the MQT coherence be-
tween 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 them-
selves differently in space, involved a very low coupling
on account of the small spatial overlap between the sin-
gle component densities in the different states [8]. 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 [9] e.g. in the sympa-
thetic 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 [10]. Such transitions could be
accessed by varying the scattering length of the boson-
fermion interaction [11].
We consider NB atomic bosons confined in a spheri-
cally symmetric harmonic trap (of frequency ωT ) inter-
acting with a much larger system of atomic fermions.
For simplicity we assume the fermions to occupy an in-
finite volume. The Hamiltonian of the bosons is de-
scribed by the standard Gross-Pitaevskii (GP) form [1],
i.e., with inter-particle interactions described by a con-
tact potential (∝ λBBδ(r − r
′)), which we choose to be
repulsive (λBB > 0) We assume that the interaction of
bosons with fermions is also contact-like, contributing
λBF |ΨB|
2|ΨF |
2 to the Hamiltonian density, where λBF
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 sys-
tem of bosons described by the functional
S = SBEC +Tr log
~∂τ −
− µF + λBF |ΨB|
where S0 is the action of the bosons alone, SBEC =
d3rΨ̇BΨ
B − HBEC), and the second term is
a contribution due to the interaction of bosons with
fermions; µF 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 inter-
ested in the dynamics of the slow breathing mode of the
BEC Ψ0B, which can be treated in the self-similar density
http://arxiv.org/abs/0704.0650v1
approximation. This dynamics describes the longitudinal
expansions (and contractions) of the condensate. Finite
size effects such as the appearance of a non-vanishing ex-
citation energy (gap) can decouple this mode from other
excitation modes. Hence, Ψ0B peaks at small frequen-
cies (ω) and small wavevectors (q), giving a Ψ0B that is
a slowly varying function of spacial and temporal coor-
dinates. In such case the Tr log[...] in Eq. (1) can be
evaluated within the Thomas-Fermi approximation. A
straightforward zero-temperature calculation yields
δTr log[...]
λBF |ΨB(r)|
ΨB, (2)
where kF is the Fermi wavevector. Eq. (2) represents an
additional term in the Gross-Pitaevskii (GP) equation,
δSGP /δΨ
B = 0, resulting from interaction with fermions.
In order to analyze the physical meaning of Eq. (2) let
us expand it in powers of ΨB. The first nontrivial con-
tribution is a term −2λ′|ΨB|
2ΨB, λ
′ = (λ2BF k
F /4π
2µF ),
which corresponds to the attraction between bosons me-
diated by interaction with fermions. For nonzero, but
small ω and q there is an additional term (of the or-
der of λ2BF ) related to the dissipation of the condensate
due to the Landau damping, as we discuss below. The
next order yields η|ΨB|
4ΨB, η = (k
BF /8π
2µ2F ). Un-
like 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 λBF ) prove to
be unimportant as can be verified directly from Eq. (2).
Therefore we will replace the potential energy contribu-
tion in GP equation given by Eq. (2) by the two terms
discussed above [12].
To analyze the dynamics of the slow (breathing) mode
described by the Hamiltonian
|∇ΨB|
2 (3)
(λBB − λ
′)|ΨB|
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
Ψ0B(r) =
4 (xR0)
2(xR0)2
, (4)
parameterized by a dimensionless parameter x that char-
acterizes the BEC’s spatial width in units of the zero
point motion amplitude R0 = (~/2mBωT )
1/2. Substitu-
tion of this wavefunction into Eq. (5) yields the following
dependence of the ground-state energy E0 on x:
E0(x) =
3NB~ωT
, (5)
where α = NB(λ
′ − λBB)/[(2π)
3/2R30~ωT ] and β =
4N2Bη/(3
5/2π3R60~ωT ). For positive but relatively small
FIG. 1: Density profiles of the bosons (solid lines 1 and 2) and
corresponding density profiles of the fermions (dashed lines
1’ and 2’) in two metastable states: (1) with fermions having
zero density at the center of the trap (“separated phase”) and
(2) with nonzero density of fermions (“mixed phase”). The
dotted line represents schematically an effective potential for
the breathing mode of the bosons.
α, i.e., for α < αcr = 32(2/5)
1/4/15 ≃ 1.69, E0 may de-
velop two competing minima, depending on the value of
the β-parameter. The energy barrier separating the min-
ima 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
−∞ at x → 0. The 1/x6 term stabilizes the system: for
small x this term rapidly increases, giving rise to another
minimum of E0(x), now due to the competition between
the last two terms in the rhs of Eq. (5).
At certain values of α and β the two minima of E0(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 tunnel-
ing transition between the two states. Such mechanism
has been suggested to be the dominant decay process for
condensates with attractive interactions between parti-
cles [5]. The tunneling corresponds to the low energy ex-
citations of the breathing mode, described by the wave-
function in Eq. (4). It has been shown in [5] that by
accounting for the superfluid motion of the condensate
(which can be done by introducing a phase-factor eiφ for
the wavefunction in Eq. (4) and requiring the superfluid
velocity vs = (~/mB)∇φ to satisfy the continuity equa-
tion) one obtains an effective action for the breathing
mode of the condensate
S0[x(τ)] =
+ E0(x)
, (6)
where E0(x) is given by Eq. (5) and m0 = 3mBNBR
Thus the dynamics of the ground state wavefunction of
the condensate is that of a quantum particle of mass m0
moving in the potential E0(x).
A direct analysis of the Shrodinger equation corre-
sponding 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, δH/δΨB =
EΨB, where H is given by Eq. (3). The latter approach
also serves as an independent justification of the varia-
tional method and confirms that macroscopic quantum
tunneling, QMT, is the mechanism that causes the tran-
sition between the two states of the condensate. Upon
substitution ΨB ∼ φ/r, the time-independent GP equa-
tion can be cast in the form
φ = µφ, (7)
where the φ(x)-function is normalized to unity, a =
(π/2)1/2α, b = 35/2πβ/16, and x = r/R0, µ = E/~ωT .
We find the ground state numerically by replacing the
rhs of Eq. (7) by −∂τφ and propagating φ in the imag-
inary time τ until it converges to the ground state φ0
(or Ψ0B). We then evaluate the ground state energy ac-
cording 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/NB)
d3r|Ψ0B(r)|
2, as a function of those same pa-
rameters. For a < acr = 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 acr corresponds
to the value of αcr = 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 conjec-
ture (e.g., Eq. (6)) of macroscopic quantum tunneling
between the two local energy minima.
The value of the tunneling matrix element ∆ between
two local “ground” states ǫ1 and ǫ2 can be deduced by
fitting the calculated energy curves in Fig. 1 with the
standard expression [13], ǫ = (ǫ1+ ǫ2)/2− [(ǫ1− ǫ2)
∆2]1/2 and assuming that in the vicinity of the point
of crossover both ǫ1 and ǫ2 are linear functions of pa-
rameter b. For a = 1.81 one finds ∆ ∼ 10−4 × ~ωT ,
while for a = 1.82, ∆ ∼ 10−2 × ~ωT . Assuming that
the ground state wavefunctions have Gaussian shape,
∼ R−3
exp(−r2/R2
), from Fig. 2 one finds
that R̄ = (R1 + R2)/2 ≃ 0.85R0 for both a = 1.81 and
a = 1.82, and δR = |R1 − R2| ≃ 0.09R0 for a = 1.81
and δR ≃ 0.03R0 for a = 1.82. For a typical value
of the trapping frequency νT = 10
2Hz (ωT = 2πνT ),
the two tunneling rates are ∆1.81/~ = 10
−2 s−1 and
∆1.82/~ = 10
2 s−1. Since the value of R0 for most
trapped atomic BEC’s is of the order of a few microns,
FIG. 2: (a) Dependence of the ground state energy of the BEC
(per particle, in units of ~ωT ) as a function of parameters a
and b; (b) Dispersion of the ground state spatial extent as a
function of the same parameters.
the difference between the radii of the two condensate
states δR 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 cos2 (∆ t/~)? The
fermions not only provide the BEC with the effective in-
teraction, 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 perturba-
tive expansion of the Tr log [...] term in Eq. (1):
(2π)3
χ0(q, ω)|ρB(ω,q)|
2. (8)
Here ρB(q, ω) is the Fourier transform of ρB(r, t) and
χ0 is the response function of the non-interacting
fermions. In the small frequency domain χ0 =
(1/4π)[~2k3F /(πµF ) + m
F |ω|/(~
2q)]. The frequency-
independent part of χ0 has already been incorporated
in the effective interaction between bosons, i.e., λ′|ΨB|
term in Eq. (3). The second term in χ0 is responsible
for damping. To quantify its role we employ a two-
state approximation in describing the tunneling dynam-
ics. In this representation the tunneling is described by
the Hamiltonian Htun = ∆σ̂x, where σ̂x is a Pauli ma-
trix with non-zero off-diagonal elements, and the posi-
tion operator, i.e. the spatial width of the ground-state
BEC wavefunction, is given by R̂ = R̄+(δR/2)σ̂z, where
σ̂z is the diagonal Pauli matrix (with ±1 along the di-
agonal). The dissipative part of the action for Htun
can be derived from Eq. (8) by substituting a Gaus-
sian ansatz, ρB(r, t) = NB/[π
3/2R3(t)] exp [−r2/R2(t)],
where R(t) = R̄ + (δR/2)σz(t), σz = ±1, into Eq. (8).
For δR ≪ R̄ one obtains
Sdiss = γ~
dτdτ ′σz(τ)σz(τ
′)(τ − τ ′)−2, (9)
where γ = N2Bλ
2/[2(2π~)4R̄4]. Eq. (9), to-
gether with Htun defined above, describes dissipative dy-
namics of a two-state system. Such dynamics has been
extensively studied in connection with macroscopic quan-
tum tunneling of a superconducting phase in Josephson
junctions, and is known to depend critically on the value
the parameter γ. Specifically, for γ > 1 the two-state os-
cillation is always overdamped and at zero temperature
it exhibits localization as a result of quantum fluctua-
tions [14]. It is therefore instructive to evaluate γ for our
situation. For estimates we consider an atomic mixture
of 23Na (bosons) and 40K (fermions), which have natural
scattering lengthes aBB ≃ 1nm (λBB = 4π~
2aBB/mB)
and aBF ≃ 4nm (λBF = 2π~
2aBF [(1/mB) + (1/mF )]).
For these data we obtain a critical value of N crB ≃ 12400
(again for νT = 10
2Hz) and the fermion density ncrF ≃
7.4×1015cm−3. Then, for a = 1.81 we obtain γ1.81 ≃ 1.1,
which corresponds to the localized case (at T = 0),
whereas for a = 1.82 one gets γ1.82 ≃ 0.1. In the high
temperature limit (for kBT > ∆) the relaxation rate Γ
can be expressed in terms of γ as ~Γ = πγkBT [14],
and therefore coherent (underdamped) oscillations can
be observed for T ≪ ∆1.82/(γ1.82kB) = 0.5nK. The
situation can be improved, however, if one utilizes a Fes-
hbach resonance [4] to increase the aBF scattering length.
For example, for aBF = 80nm one finds N
B ≃ 25 and
ncrF ≃ 2.6 × 10
11cm−3, and γ1.82 ≃ 2.5 × 10
−4. For
such parameters coherent oscillations can be observed for
T ≪ 0.2µK, which is easily observable. A low particle
number also reduces the uncertainty of an atomic count-
ing measurement that can be carried out in the time-of-
flight procedure [15].
In summary we argue that a trapped boson-fermion
mixture can exhibit MQT tunneling and coherent oscil-
lations. Our studies indicate that MQT can be observed
in 23Na and 40K atomic mixtures of sufficiently low tem-
peratures.
We thank M. Boshier and S. A. Gurvitz for valuable
discussions. The work is supported by the US DOE.
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