Ramsey fringes formation during excitation of topological modes in a
  Bose-Einstein condensate
Ramsey fringes formation during excitation of
topological modes in a Bose-Einstein
condensate
E. R. F. Ramos a, L. Sanz a, V. I. Yukalov b and V. S. Bagnato a
aInstituto de F́ısica de São Carlos, Universidade de São Paulo,
Caixa Postal 369, 13560-970, São Carlos-SP Brazil
bBogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research
Dubna 141980, Russia
Abstract
The Ramsey fringes formation during the excitation of topological coherent modes
of a Bose-Einstein condensate by an external modulating field is considered. The
Ramsey fringes appear when a series of pulses of the excitation field is applied. In
both Rabi and Ramsey interrogations, there is a shift of the population maximum
transfer due to the strong non-linearity present in the system. It is found that the
Ramsey pattern itself retains information about the accumulated relative phase
between both ground and excited coherent modes.
Key words: Bose-Einstein condensation, Ramsey fringes, Coherence, Non-linear
dynamics
PACS: 03.75.Kk, 39.20.+q, 31.15.Gy, 42.25.Kb
1 Introduction
The creation of Bose-Einstein condensates (BEC) in non-ground states, as
originally proposed by Yukalov, Yukalova, and Bagnato [1,2] is nowadays a
topic of wide interest. This method allows for the direct formation of frag-
mented nonequilibrium condensates (see the review article by Leggett [3]).
Email address: edmir@ursa.ifsc.usp.br (E. R. F. Ramos).
Preprint submitted to Elsevier 11 August 2021
http://arxiv.org/abs/0704.1953v1
One of important applications of coupling between different coherent modes of
BEC is the possibility to produce various transverse modes in the atom laser
[5,6,7,8]. Many experiments are devoted to the study of properties of trapped
BECs by coupling their collective states [9,10].
Starting with a sample of atoms Bose-condensed in the ground state of a con-
fining potential, it is possible to promote atoms from one trap level to another
by using resonance excitation [1,11]. This is done by applying an additional
weak external field, with a fixed spatial distribution, and oscillating in time
with a frequency near the transition frequency between the ground and an ex-
cited state. Previous calculations demonstrated the possibility of macroscopic
transfer between the levels, confirming the feasibility of this procedure, and
analyzed several applications [1,2,11].
In the present work, we investigate the formation of the Ramsey-like fringes,
due to the interference between the ground and a non-ground states of BEC,
excited by means of a near resonant field. We keep in mind a time-domain ver-
sion of the separated oscillatory-field method, as developed by Ramsey [13,14],
consisting of a sequence of two Rabi π/2-pulses, which are equivalent to the
oscillatory fields of the Ramsey method. Previously, the formation of Ramsey
fringes in double Bose-Einstein condensates [15] was studied. But this is for
the first time that the Ramsey patterns are obtained, when BECs states with
different quantum numbers, associated with the trap potential, are coupled.
The eventual measurement of those fringes would quantify the coherence of the
process. An experimental setup for the observation of the topological coherent
modes, through spatial distribution observation, is presently in progress in our
research group [16,17].
This paper is structured as follows. In Sec. 2, we briefly review the dynamics of
the coherent modes, based on the Gross-Pitaevskii equation for BEC [4]. Also,
we recall the idea of the resonant excitation, which makes it possible to couple
the ground and non-ground states. Section 3 is devoted to the formation of the
Ramsey-like fringes and to the possibility of their experimental observation.
In Section 4, we summarize our results.
2 Gross-Pitaevskii equation and coherent modes
At low temperatures, dilute Bose gas, as is known, is well described by the
Gross-Pitaevskii equation. This equation describes the coherent states of the
Bose system with the Hamiltonian
Ψ̂† (r, t)
+ Vho (r, t)
Ψ̂ (r, t) dr+
drdr′Ψ̂† (r, t) Ψ̂† (r′, t)V (r− r′) Ψ̂ (r′, t) Ψ̂ (r, t) , (1)
with N bosons assumed to be confined by a harmonic trap potential, Vho (r, t).
The Ψ̂ (r, t) (Ψ̂† (r, t)) are the boson field operators that annihilate (create)
an atom at position r and V (r− r′) is a two-body interaction due to atomic
collisions. In a dilute cold gas, the most relevant scattering process is associated
with elastic binary collisions at low energy [3,4], giving the effective interaction
potential
V (r− r′) = Asδ (r− r
′) = (N − 1)
4π~2as
δ (r− r′) . (2)
Here, as is the “zero-energy” s-wave scattering length. For an effective attrac-
tive (repulsive) interaction as is negative (positive). The confining harmonic
potential is written as
Vho (r, t) =
2 + ω2yy
2 + ω2zz
where m0 is the atomic mass and ωk (k = x, y, z) are the trap oscillation
frequencies along each axis.
The topological coherent modes are the solutions to the stationary Gross-
Pitaevskii equation. The nonlinear coherent modes form an overcomplete ba-
sis and are normalized (|ϕj|
= 1). The eigenvalue problem for stationary
functions is given by [11]
Ĥ [ϕj]ϕj = Ejϕj (4)
where
Ĥ [ϕj ] = −
+ Vho (r, t) + As |ϕj|
follows from the Hamiltonian (1).
The resonant pumping makes it possible a coherent transfer of condensed
atoms between the collective levels of atoms in the harmonic trap, as is de-
scribed in Ref. [11]. At initial time, N condensed atoms are assumed to be in
the ground state of the trap with a frequency ω0. The aim is to transfer the
atoms to a non-ground state (labelled as p). The energy difference of the level-
p, relative to the ground state, is ~ (ωp − ω0). Following Yukalov et al. [11],
this transfer can be obtained through the action of an external oscillatory field
given by
Vp (r, t) = V (r) cosωt . (6)
The corresponding non-linear Schrödinger equation, associated with the Hamil-
tonian (5), writes as
∂φ (r, t)
Ĥ + Vp (r, t)
φ (r, t) . (7)
The modulating field is called external in the sense that it is not a part of the
harmonic trap setup, but rather the field Vp (r, t) is a perturbation in Eq.(7).
The solution φ (r, t), can be expressed as a sum over the coherent modes:
φ (r, t) =
n cn (t)ϕn (r, t). Although one could consider a more general form
of the external field for transferring atoms between an arbitrary pair of collec-
tive levels [11], it is sufficient to work with the potential (6) in order to create
the Ramsey interference fringes.
The functions c0 (t), associated with ground state, and cp (t), ascribed to an
excited state, determine the behavior of the populations in each mode
n0 (t) = |c0 (t)|
, np (t) = |cp (t)|
. (8)
There are some conditions on the involved physical parameters in order to
observe a macroscopic population of a non-ground state (np 6= 0): First, it is
required that the resonance condition between the external field of frequency
ω and the frequency associated with the level transition, ωp0 = ωp−ω0, be ful-
filled. Also, the detuning ∆ω = ω−ωp0 should be small enough, |∆ω/ωp0| ≪ 1.
At the same time, one should remember that not solely the external coupling
filed, but also the interatomic collisions cause atomic transitions between the
modes. In order to quantify both effects, it is convenient to define the inter-
action intensity α, associated with interatomic collisions, and the transition
amplitude β of the coupling field as
αm,k =
|ϕm (r)|
2 |ϕk (r)|
− |ϕm (r)|
(r)Vp (r)ϕp (r) dr. (9)
For simplicity, we set α ≡ α0,p = αp,0 for the folowing analysis. These quan-
tities have dimensions of frequency (Hz). Their values are to be smaller than
the transition frequency ωp0, so that
≪ 1 ,
≪ 1. (10)
For the time dependent process, with the potential Vp, the solution of Eq.(7)
can be represented as
φ (r, t) =
cn (t)ϕn (r) exp
Ent . (11)
Another condition is that the coefficients cn (t) evolve slowly in time, when
compared to the oscillatory terms in the equation (11), so that
≪ 1. (12)
It is straightforward to obtain a system of coupled differential equations for
the coefficients c0 and cp, whose detailed derivation can be found in Ref. [1],
=αnpc0 +
βei∆ωtcp,
=αn0cp +
β∗e−i∆ωtc0. (13)
Let us mention that the form of these coupled equations resembles the equa-
tions of the Rabi two-level problem [18]. It is expected that, if we choose
the appropriate physical parameters, Rabi oscillations between the topolog-
ical modes could be observed. The solution to this system provides us with
all necessary information about the dynamics of the populations, including
both ground and non-ground states. An analytical solution of Eqs.(13) can
be obtained when |β| ≪ |α|, as in Ref. [2]. Then the population fractions are
given by
n0 =1−
, (14)
with the effective collective frequency Ω defined as
+ [α (n0 − np)−∆ω]
. (15)
It is worth emphasizing that this collective frequency is not a constant, but a
function of the instant populations n0 and np, and that the analytical solution
is available under the condition that the populations of the modes have small
changes. Thus, the population difference ∆n = |np − n0| is almost a constant.
When we are interested in a macroscopic population of an excited state, it
is necessary to solve the coupled equations (13) using a numerical procedure
based on the fourth-order Runge-Kutta method. Thus, we can deal with α
and β values not restricted to the condition |β| ≪ |α|.
In order to find the time dependent solutions for coefficients c0 (t) and cp (t),
we consider as initial conditions c0 (0) = 1 and cp (0) = 0 (all N atoms ini-
tially condensed in the ground state). In figure 1, we plot the results for the
fractional populations, associated with the non-ground and excited states and
0 10 20 30 40 50
0 10 20 30 40 50
Fig. 1. Evolution of the density of populations (n0 in dotted line, np in grey line) and
the population unbalance ∆n = |np − n0| (black line) as function of dimensionless
quantity αt for ∆ω ≈ 0 and different values of coupling parameter β: (a) β = 0.40α;
(b) β = 0.46α; (c) β ≈ 0.50α and (d) β = 0.60α.
the population difference as functions of the dimensionless parameter αt. Grey
lines show the population of the non-ground state (np = 0 at initial time) and
dotted lines show n0 (t). The evolution of the population difference, ∆n (t),
is plotted in black bold lines. For all cases, the detuning is fixed so that
∆ω ≈ 0, and we simulate a slight change of the external field Vp, which is
characterized by the coupling parameter β defined in Eq.(9). From the previ-
ous works [19,11,2], we know that there are critical effects associated with the
coupled system given by Eqs.(13), which, however, we shall not consider here.
Since it is possible to transfer populations between two different topological
states, under well defined conditions, prescribing when this process is efficient,
we are in a position to analyze what happens if we manipulate the excitation
time domain in order to detect Ramsey-like interference fringes as the oscilla-
tions of the fractional populations.
3 π/2 and π- pulses and the formation of the Ramsey-like fringes
In this section, we demonstrate the formation of the Ramsey-like fringes for
the case of two topological modes coupled by an external resonant field. Given
a certain value of β, we define the maximum of the atom population that is
transferred from the ground to an excited state. It is also possible to estimate
the necessary time for the coupling to be switched on. When a π pulse is
applied, the time of the coupling, t1, is sufficient for the performance of a
complete Rabi oscillation. If the coupling is switched off at the time, when
the half of the maximum population is transferred, the applied pulse is the so-
called Rabi π/2-pulse. We show below that Ramsey fringes are obtained when
two Rabi π/2-pulses are applied with a time interval τ between them. The
latter procedure is similar to that one accomplished for the coupled hyperfine
levels of Rubidium [20], and the formation of the fringes confirms the existence
of a relative phase between both topological modes.
In figure 2, we plot our results for the fractional population density of the
ground state, n0, considering three pulse configurations, mentioned above, as
functions of the detuning ∆ω. The normalized quantity n0 gives us information
about the atomic population in the mode and an indirect quantification of the
visibility for the imaging process. The total time of the simulations is equal in
all three cases. The procedure of obtaining the patterns implies the solution
of system (13) for a fixed value of β, which means a fixed value of the spatial
amplitude function in Vp, allowing for the variation of the detuning ∆ω.
In figure 2, black lines show the expected behavior of n0, when equivalent Rabi
pulses are applied. Both, π (bold line) and π/2 (thick line), pulses exhibit
common features. First, we observe a sole peak, with a maximum at ∆ω 6= 0.
The shift of the maximum of the resonant condition is associated with the
contribution from the nonlinear terms of the Hamiltonian (5). Second, the
maximual value of n0 depends on the number of atoms transferred between
the coupled states, which, in turn, depends on the coupling time t1. This is
related to the halfwidth Γ ≈ 1
. If we compare both black lines in figure 2, we
note that Γπ ∼ 1.44 and Γπ/2 ∼ 2.76 ≈ 2Γπ.
The possibility of the formation of Ramsey fringes was first suggested in
Ref. [11]. Our numerical calculations demonstrate that the fringes are really
obtained when two π/2 pulses are applied separately. In this way, given a fixed
-6 -4 -2 0 2 4 6
Fig. 2. The fractional population of a non-ground state, np, as a function of detuning
∆ω for three different kinds of pulse configurations. For all cases β = 0.4α. Grey line:
Ramsey fringes due to the application of two π/2 pulses with an interval τ = 8t1;
Bold black line: a π Rabi pulse; Thick black line: π/2 Rabi pulse.
number of condensed atoms N , approximately half of the population is trans-
ferred from the ground to an excited state during the first π/2 pulse, evolving
freely when the coupling is switched off, and then, a second pulse concludes
the excitation. The fractional population after this process, as a function of
the detuning ∆ω, is plotted in figure 2 with the grey line. The process simu-
lates the effect of the application of two π/2 Rabi pulses separated by τ = 8t1.
The maximal value for np is found at the same frequency as in the configu-
ration of a single pulse, which confirms our previous conclusion that the shift
in the frequency is a hallmark of the nonlinear effects due to elastic two-body
collisions.
In figure 3, we plot the behavior of the fractional population as a function of β,
considering the same pulse configuration as above, with an initial π/2 pulse,
an interval τ = 4t1, and a second π/2 pulse. If the coupling amplitude is weak,
the Ramsey pattern loses its visibility. Increasing β improves visibility, but at
the same time causes power shifts, represented by the fringes displacement.
The number of the Ramsey fringes, contained within the spectral width, is
strongly dependent on the time interval τ between the applied π/2- pulses.
This feature is shown in figure 4, where four different situations are considered.
We observe that the absolute maximal values of np and its position, as a
function of ∆ω, are not connected with the changes of τ . What does depend
-4 -2 0 2 4
Fig. 3. Ramsey fringes of the fractional populatuion np as a function of ∆ω, after the
application of two π/2- pulses separated by τ = 4t1, with t1 being the time needed
for the excitation of the half of the population from the ground to a non-ground
state. The lines correspond to the increasing values of β. Light grey line: β = 0.1α;
Grey line: β = 0.2α; Black line: β = 0.3α
on τ is the number of fringes, which increases as τ increases. If τ is equal to the
coupling time t1, two additional peaks appear at the both sides of the main
peak. For τ = 2t1, we obtain two peaks at each side, and so on, as it can be
seen comparing the four plots in figure 4. The appearance of these auxiliary
peaks is the hallmark for the accumulation of a relative phase between both
topological states. During the time τ , when the coupling is switched off, the
dynamics of the whole system (ground + non-ground state) is associated with
the non-linear term in the Hamiltonian (5). This free dynamics determines the
accumulated relative phase.
-4 -2 0 2 4-4 -2 0 2 4
  = 2t
  = t
  = 3t
  = 8t
Fig. 4. Ramsey fringes of the fractional population np as a function of ∆ω, after the
application of the Ramsey pulse configuration for different τ values. The gray line
in all plots corresponds to the Rabi π/2-pulse.
4 Conclusions
In this work, we use numerical calculations for solving the system of coupled
equations describing the resonant excitation of two coherent non-linear modes
of BEC. Our results, obtained by means of the fourth-order Runge-Kutta
method, give us an insight into the behavior of the fractional mode popula-
tions of non-ground collective atomic states in a harmonic trap. The principal
novelty of the present paper is the investigation of the system response to
different coupling configurations and the demonstration of the appearance of
the Ramsey-like fringes.
The formation of the Ramsey-like fringes is a signature of the actual coherent
character of topological modes in the studied nonlinear system. In both, Rabi
and Ramsey pulse configurations, there is a shift of the maximum population
transfer due to the strong effect of non-linearity of the system. The Ramsey
pattern itself contains information on the accumulated relative phase, and
the number of secondary peaks is proportional to the time τ , as defined in
Section 3.
As possible extension of this work, it would be interesting to consider the
influence on the Ramsay fringes of the trap geometry and of different external
pumping fields Vp (r, t). Other possibilities could be related to the manipilation
of the scattering length via the Feshbach resonance techniques and to the
effects of changing the total number of atoms, which affects the interatomic
intensity α.
Acknowledgments
The authors wish to thank E. P. Yukalova for her important contribution at the
early stage of this work. Special thanks are to E. A. L. Henn and K. M. F. Mag-
alhães for helpful discussions. This work was supported by Fapesp (Fundação
de Amparo à Pesquisa do Estado de São Paulo), CAPES (Coordenação de
Aperfeiçoamento de Pessoal de Ńıvel Superior), and by CNPq (Conselho Na-
cional de Desenvolvimento Cient́ıfico e Tecnológico).
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	Introduction
	Gross-Pitaevskii equation and coherent modes
	/2 and - pulses and the formation of the Ramsey-like fringes
	Conclusions
	References
