Download Nonlinear matter wave optics

Nonlinear matter wave optics
Bernhard Michl
Author, Universität Mainz
Immanuel Bloch
Supervisor, Universität Mainz
(Dated: 26 July 2006)
Solitons, an effect of nonlinear matter wave optics, are nonspreading localized wave packets. Bright
matter wave solitons can be observed if the nonlinear dynamics compensates the linear dispersion
of matter waves. Nonlinearity is given by the interaction between the particles in a Bose-Einstein
condensate, the dispersion acts on every material particle even in vacuum. These effects are shown
in two examples: In the experiment of Khaykovich et al. (2002) solitons with 7 Li are produced by
controlling the interactions between the particles. Eiermann et al. (2003) achieves the same with
87
Rb by controlling the dispersion.
INTRODUCTION
Solitons are encountered in many different fields, such
as oceanography (Tsunamis), biology (signal transmission in nerves) and physics. J.S. Russell described 1834
for the first time the formation of a soliton in a narrow water channel. This water wave didn’t change its
shape for a long distance. In physics solitons are found
in systems with either photons and matter waves. The
optical solitons shall be used in fiberglasses for information transport. Because of the nonspreading high transfer rates over long distances can be achieved (5-Gbit/s
over 15000km [1]). Another aspect of solitons is that the
soliton is a very stable wave packet form. They keep
their shape even if perturbations occur, which is always
the case when talking about real systems. Therefore I
shortly explain the effects that lead to the formation of
optical solitons in the second part. Afterwards we discuss
material systems. A discussion of Bose-Einstein condensation (BEC) is the basic for the experiments of matter
wave solitons. In the third part the experiment and the
theoretical background is explained.
nonlinear refractive index.
n = n0 + nnl |E|2
n is in the spatial term of the phase Φ = ω0 t − nk0 x and
therefore the phase shifts dependent of the light intensity.
In a temporarily gaussian wave packet (with hypothetical
only one frequency ω0 ) the phase in the center, where
intensity is highest, changes fast. At the front and at the
end there is almost no change in phase. So on the one
side the oscillation gets more frequent, on the other side
lower frequencies are generated. Quantitative one must
have a look at the deviation of the momentary frequency
ω(t) from ω0 (t).
∆ω(t) = ω(t) − ω0 (t) =
∂Φnl
∂t
Φnl is the nonlinear part of the phase proportional to the
gaussian profile. The derivation of the gaussian profile
is shown in figure 1. Lower Frequencies are at the front
(chronological earlier, small value of time t), higher at
the end (chronological later, high value of t).
NONLINEAR OPTICS
Nonlinearity in optics occurs when the light waves have
high intensities, so these problems can be investigated
since the invention of the laser in the 1960s. When a
light wave propagates through matter at low intensity
the electrical field excites the electrons to harmonic oscillations. If the amplitude gets too large, anharmonic
oscillations appear and the emitted wave can consist of
other frequencies than the incident wave. Here the relation between the polarisation and the electrical field
strength is nonlinear. One nonlinear effect is the self
phase modulation (SPM). This nonlinearity is important
when optical solitons form. From the relation between
polarisation and electrical field strength one can find a
FIG. 1: Gaussian Profile with derivation of the nonlinear
phase [2]
Now the well known effect of anomal group velocity
2
dispersion (GVD) comes into play. Here no new frequencies arise, but the existing lower frequencies move to the
end, the higher to the front. One can see, that the SPM
and the anomal GVD have the capability to compensate
each other. Quantitative the following differential equation describes the dynamics:
∂2
∂
i A(x, t) = β 2 − γ|A(x, t)|2 A(x, t)
∂x
∂t
β < 0 belongs to anomal GVD, γ > 0 to the SPM. The
soliton solution A(x, t) of the equation have the characteristic shape of a secans hyperbolicus [1].
BEC
No we want to focus on systems with atoms and discuss under which conditions solitons appear. The BoseEinstein condensation is the basis for this, because in a
BEC all atoms have the same quantum mechanical wavefunction and the whole BEC can be described by a macroscopic wavefunction Ψ. BEC as a new state of matter
was theoretical predicted in 1925 by Albert Einstein. He
translated workings on photons of Satyendra Nath Bose,
an Indian physicist and transferred the principles to matter. BEC arises when the wavefunctions of the atoms
overlap. A measure for the width of the wave
packets is
h
,
the thermal DeBroglie wavelength λdB = √2πmk
T
B
a measure for the distance between the atoms is n−1/3 ,
where n is the particle density (one thinks of a cubic crystal lattice with volume V and number of atoms N, the
edge length is V 1/3 and the number of atoms on the edge
N 1/3 , n is length divided by number). So an overlap of
wavefunctions means λdB ≈ n−1/3 . Now the result of the
theoretical statistical physics seems feasible, it says that
BEC occurs when
is detuned from the optical transition ω21 in the atoms.
For the potential we get in quantum mechanics
V ∝
I(~r)
δ
.
I(~r) is the intensity of the laser beam, δ is the detuning,
that has to be much larger than the Rabi frequency. For
δ < 0 a maximum of intensity is a minimum of the potential for the atoms [3]. These trapped atoms can be
condensed by evaporative cooling. Therefore the potential walls are made lower, fast atoms can leave the trap
and the others rethermalize at a lower temperature. For
every magnitude of lost atoms, Ω increases approximately
two magnitudes. 1995 the first experimental evidence of
BEC is found by Cornell and Wiemann. Ketterle made
advanced experiments with BEC. All three got the nobel
prize 2001.
NONLINEAR MATTER WAVE OPTICS
When discussing nonlinear matter wave optics first of
all we have to clarify where the nonlinearity comes from.
Therefore we must have a look at quantum mechanical
scattering. The scattering potential is given by the attractive van der Waals potential that always acts between
particles and a term due to the (zentrifugal barrier). This
term comes from the seperation between radial and angle
problem, a similar procedure like in the known hydrogen
problem. On the right of figure 2 you see that a particle
of low energy is reflected by the barrier and the probability of finding stays the same.
Ω := nλ3dB ≥ ζ(3/2) ≈ 2, 613
Ω is called the phase space density, ζ(s) is the Riemann
zeta function. Our aim is a condensate of rubidium
atoms. We can get them from rubidium chromate at
a temperature of 1000 K. The problem is that Ω of the
atoms in the oven is approximately 10−17 . 10 orders of
magnitued are achieved by laser cooling and trapping
the atoms in a magneto optical trap. The problem of the
MOT is that it functions with a resonant coupling of photons and the atoms. Because of the process of absorbing
and emitting photons the density and the temperature is
limited. The remaining 7 magnitudes can only be overcome in a dark trap, i. e. a magnetical or a optical dipole
trap. I will not dwell on the magnetical trap, because
the dipole trap will be of more importance for the experiments introduced in the next part. The dipole trap
is based on a non-resonant laser with frequency ωL that
FIG. 2: a) s-wave scattering, van der Waals potential; b)
scattering with l=2, effective potential (dashed line) [4]
The contribution that is important is the l=0 Term, the
s-wave scattering. To see the basic effects the potential
can be approximated by a simple negative box potential.
When solving this problem one finds solutions dependent
of the depth V0 and width R0 of the box like these in
figure 3.
The negative potential can therefore have a repulsive
effect. An important aspect here is the boundary condition of the continuity at the box edge. The good
3
interaction. The dispersion term here is the kinetic term.
One result of elematary quantum mechanic is that the
mass m controls the dispersion. Heavy particles show
slow dispersion, light ones a fast dispersion in vacuum.
FIG. 3: Solutions of the box potential [5]
result is, that in this simple regime only one parameter rules the whole scattering behaviour, the scattering
0)
length a = − limk→0 tan(δ
. It is defined with the phase
k
shift δ0 shown in figure 3 in the limit of low energy, i.e.
low wave vectors k. a < 0 corresponds to a attractive potential, a > 0 to a repulsive one. To describe the process
in an easy way the van der Waals potential between two
particles at r1 and r2 is substituted by a delta potential
VK (r1 − r2 ) ∝ a · δ(r1 − r2 ) that has the same scattering
length, i.e. the same physical properties. Furthermore
the potential of all particles acting on one particle is approximated with a mean-field-description.
Z
V (r) = n(r′ )VK (r′ − r)dr′
FIG. 4: Black soliton [6]
Now this equation has two solitonic solutions: Dark
solitons arise for different signs of dispersion and nonlinearity (figure 4 shows a black one with zero density in the
center), bright solitons for equal sign of dispersion and
nonlinearity (figure 5). In the following only bright solitons are discussed, as they are of more interest, because
the whole BEC has the shape of a soliton.
Inserting the so called contact potential VK in this equation, the structure gets very simple.
V (r) ∝ a · n(r) = a|Ψ|2
The particle density n is substitued by the squared norm
of the macroscopic wave function Ψ. Here Ψ is not
the function of a single particle, but of all the particles
together. This description is possible in a BEC. Now
we found a potential term that describes the interaction between particles well. In the Schrödinger equation
this term has to be considered, the resulting nonlinear
Schrödinger equation is called Gross-Pitaevskii differential equation:
∂
h̄2 ∂ 2
2
ih̄ Ψ(x, t) = −
Ψ(x, t)
+
a
·
C
·
|Ψ|
∂t
2m ∂x2
C is an abbreviation of some constants, Ψ is again the
macroscopic wave function. The structure of the equation is the same as in the equation we discussed for the
optical problem. The nonlinearity is given by the atom
FIG. 5: Bright soliton [6]
Solitons by controlling the nonlinearity
In a first experiment we will see how one can derive
bright solitons with 7 Li. Considering that the physical
mass m has always a positive value, for bright solitons the
sign of the nonlinearity has to be also negative. Therefore
a negative scattering length is needed. The question is if
one can control the interaction between the atoms so it
gets attractive. The answer give the so called Feshbach
4
resonances. In figure 6 you see that applying a magnetic
field on the atoms changes the scattering length.
FIG. 6: Feshbach Resoncance [7]
At some point the scattering length diverges and the
experimenter will find a maximum in the loss rate in the
trap (small graph in figure 6). This phenomenon is quite
complicated, but one can understand it roughly when
going back to the problem of the potential box with an
incident wave. The solution is dependent on the energy
levels in the box. An energy level at the vacuum level (energy E=0) leads to a diverging scattering length. With
the magnetic field these energy levels can be varied so
that a Feshbach resoncane occurs. In an experiment by
Khaykovich et al. (2002) bright solitons were generated
by using Feshbach resonances. In figure 7 the situation
is shown for an ideal gas (a=0) and an attractive interaction (a < 0). For the ideal gas the normal dispersion is
observed. With the attractive interaction a wave packet
is formed that shows almost no changes in shape.
FIG. 7: A: ideal BEC gas; B: Soliton [8]
Solitons by controlling the dispersion
Now a second experiment with 87 Rb is presented. The
scattering length for 87 Rb is positive and no Feshbach
resonance is used. For bright solitons and equal sign is
needed, therefore the mass has to be negative. Because
the physical mass is never negative, an effective mass
formalism has to be introduced. The formalism is very
similar to that in solid state physics, where the electrons
move in the periodic potential of the atomic nucleus [9].
In our case a periodic potential is generated very easily
with two counterpropagating laser beams. The resulting
standing wave has a spatial periodicity in the intensity
and regarding the equation of the dipole trap we had
before this leeds to a periodic potential. This potential
is inserted in the Gross-Pitaevskii differential equation.
For small periodic depths the condensate wave function
is only modulated by the high frequent periodic potential
(figure 8).
FIG. 8: Ψ(x) in the spatial space [10]
Therefore one can find an equation for the envelope
function f(x).
∂
h̄2 ∂ 2
2
ih̄ f (x, t) = −
+ a · αkorr · |f | f (x, t)
∂t
2mef f ∂x2
αkorr is a correction term. The effective mass is given by
mef f (k) = h̄
2
∂ 2 E(k)
∂k 2
−1
.
Like in the solid state physics energy bands E(k) occur
that are dependent on the quasi momentum k. Its called
5
quasi because it is also periodic, not like a normal momentum. In figure 9 the energy bands, group velocities vg (k) = h̄1 ∂E(k)
and the effective mass are shown.
∂k
h̄2 k2
Erec = 2mL is the recoil energy of the photons with kL
the wave vector of the laser beam.
FIG. 11: Bright soliton [12]
CONCLUSION
FIG. 9: a) energy bands with BEC in the momentum space;
b) group velocity; c) effective mass [11]
One can see that there is a region of constant negative effective mass at the edge of the brillouin zone (yellow marked). When switching on the potential the condensate is prepared in the middle of the brillouin zone
with constant positive effective mass (green region). For
preparing the BEC at the edge a force has to be applied.
That is realized by detuning the two laser beams. This
leads to a moving potential V (x) = V0 sin2 (kL x − ∆ω
2 t).
The velocity is determined by the detuning ∆ω. Accelarating the potential leads to a vis inertiae on the condensate in the framework of the resting lattice potential
(figure 10).
FIG. 10: Preparation of the BEC at the band edge [12]
The experimental results of this method used in 2003
by Eiermann et al. are shown in figure 11. First the
soliton stabilises, some atoms are repelled. For 45ms a
wave packet of 250 atoms could be observed.
Both experiments have advantages and disadvanteges.
Using the method of Khaykovich you first of all have to
find a Feshbach resonance which can be difficult for some
atoms. Furthermore the number of atoms in the attractive regime is limited because the system collapses if too
many atoms are used. The observation time was limited
by an expulsive potential caused by the magnetic field.
In the build up of Eiermann every atom can be used and
many phenomenons of solid state physics can be investigated. Because the group velocity vanishes at the edge
of the brillouin zone you get standing solitons. But the
number of atoms is also limited because the interaction
is only affected by the atom density.
Future Applications of atomic solitons could eventually
be in coherent atom optics, atom interferometry and
atom transport.
[1] Bergmann/Schaefer, Optik (de Gruyter, 2004), 10th ed.
[2] S. Leinss, Solitonen, Universität Konstanz, Speech
(2006).
[3] H. J. Metcalf and P. van der Straten, Laser Cooling and
Trapping (Springer, 1999).
[4] A. Marte, Ph.D. thesis, TU München (2003).
[5] T. Mayer-Kuckuk, Kernphysik (Teubner, 2002), 7th ed.
[6] B. Eiermann, Ph.D. thesis, Universität Konstanz (2004).
[7] K. Strecker et al., New Journal of Physics 5, 73.1 (2003).
[8] L. Khaykovich et al., Sciencemag 296, 1290 (2002).
[9] N. W. Ashcroft and N. D. Mermin, Solid State Physics
(Oldenbourg, 2001).
[10] P. Treutlein, Master’s thesis, Universität Konstanz
(2002).
[11] M. Taglieber, Master’s thesis, Universität Konstanz
(2003).
[12] B. Eiermann et al., Physical Review Letters 9, 23 (2003).