UNIVERSIT ` A DEGLI STUDI DI MILANO Facolt` a di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Fisica TOWARD THE EXPERIMENTAL REALIZATION OF QUANTUM COLLECTIVE ATOMIC RECOIL LASING WITH BOSE-EINSTEIN CONDENSATES IN A RING CAVITY Relatore Interno : Prof. Nicola Piovella Relatore Esterno : Prof. Philippe Courteille PACS: 42.50.Fx, 42.50.Vk Tesi di Laurea di Valeria Brizzolara matr. 720899 Anno Accademico 2007 - 2008
Transcript
UNIVERSITA DEGLI STUDI DI MILANO
Facolta di Scienze Matematiche, Fisiche e Naturali
The equations for θj and for pθjderive from the Hamiltonian:
H =N∑
j=1
[p2
θj
2ρ− iρ(Aeiθj − A∗e−iθj)
]=
N∑j=1
Hj (1.17)
which satisfy the following canonical equations:
dθj
dt=
∂H
∂pθj
dpθj
dt= −∂H
∂θj
The N atoms are independent because the potential depends only on the self
consistent field A, assumed to be classical. Instead of solving the N Heisenberg
equations for θj and pθj we consider the Schrodinger equation for the wave function
Ψ(θ, t) which represents the statistical ensemble of the particles:
i∂Ψ(θ, t)
∂t= H1Ψ(θ, t) = − 1
2ρ
∂2Ψ
∂θ2− iρ(Aeiθ − A∗e−iθ)Ψ (1.18)
where H1 is the single-particle Hamiltonian. The quantized expression of (1.12) can
be obtained replacing the sum over the particles in the bunching factor (1/N)∑
j exp(−iθj)
by∫ |Ψ|2 exp(−iθ)dθ, where |Ψ|2 is the atomic density in the periodic system of
particledA
dt=
∫ 2π
0
|Ψ(θ, t)|2e−iθ + iδA− κA (1.19)
Equations (1.18) and (1.19) form the simplest quantum model describing semiclas-
sically the behavior of a Bose-Einstein Condensate in CARL.
Chapter 1. Collective Atomic Recoil Laser 14
Generally a BEC is realized when the atoms are so close that their wave func-
tions overlap. Below the TC of the transition the atoms occupy all the same
quantum macroscopic state, in particular because of the Heisenberg uncertainty
principle σzσpz > ~/2 and σpz ∼ ~/2L, where L is the extension of the condensate.
The quantum regime of CARL could be observed only if the momentum spread is
smaller than the recoil momentum, 2~k, which means that ~/2L ¿ 2~k or L À λ.
For a condensate with extension much larger than the optical wavelength, the mo-
mentum spread due to the Heisenberg uncertainty principle can be neglected and
the atoms can be assumed completely delocalized in a radiation wavelength, so
that Φ(θ, 0) ≈ 1/√
2π at the initial time.
So far we made the hypothesis T = 0K. The model of Eqs. (1.18) and
(1.19) can be generalized when T 6= 0K taking into account for inhomogeneous
broadening of the atomic velocities distribution (see [25]). In particular if atoms
have different momentum they will have different detuning, since δ = ω−ωp− kv.
The effect of a distribution G0(δ) (describing different classes of atoms each one
with its own detuning δ) is more conveniently included in a model where Ψ(θ, t)
is expanded in Fourier series 1:
Ψ(θ, t) =+∞∑
m=−∞cm(t)eim(θ+δt) (1.20)
The coefficients |cm(t, δ)|2 can be interpreted quantum-mechanically as the prob-
ably amplitude to find an atom with detuning δ and momentum p = m(2~k). In
fact Eq. (1.20) is the expansion of the atomic wave function on the basis of p with
eigenstates exp(imθ) and eigenvalues m(2~k). The atoms change their momentum
only by discrete steps of 2~k when they backscatter a photon from the pump to
the CARL mode or vice versa.
Inserting Eq.(1.20) into Eqs. (1.18) and (1.19), defining A = Aeiδt and general-
izing the source term of Eq. (1.19) to include all the different detunings δ weighted
by G0(δ), the motion’s equations become:
dcn
dt= −in
(n
2ρ+ δ
)cn − ρ(Acn−1 − A
∗cn+1) (1.21)
dA
dt=
+∞∑n=−∞
∫ +∞
−∞dδG0(δ)c
∗n−1cn − κA. (1.22)
The term on the right-hand side of Eq. (1.22) is the quantum expression for the
local bunching b = 〈eiθ〉 =∫
dδG0(δ)∑
n c∗n−1(δ)cn(δ) and shows that lasing is pos-
1Ψ can be expanded since θ is a periodic variable between 0 and 2π and it is an homogeneousfunction.
Chapter 1. Collective Atomic Recoil Laser 15
sible only in the presence of a coherent superposition of two adjacent momentum
states of atoms.
As in Classical CARL model it is interesting to perform a stability analysis of
Eqs. (1.21) and (1.22) of the initial equilibrium state with no field A = 0 and
all the atoms in the same momentum state n (cn = eiωnt and cm = 0 for m 6= n,
ωn = n(n/2ρ + δ) are the eigenvalues of the system). Perturbing this equilibrium
state with infinitesimal quantities in the field A and in amplitudes cm Eqs. (1.21),
(1.22) reduce to:
da1
dt= −i
(δn − 1
2ρ
)a1 + ρA (1.23)
da2
dt= −i
(δn +
1
2ρ
)a2 − ρA (1.24)
dA
dt=
∫ +∞
−∞dδG0(δ)[a1 + a2]− κA (1.25)
where δn = δ + n/ρ, a2 = cn+1eiωnt and a1 = c∗n−1e
−iωnt. Introducing the Laplace
transform in t,
f(λ) =1
2π
∫ +∞
0
dtf(t)e−iλt
and assuming a1(0) = a2(0) = 0 for a continuous beam and A(0) = A0, Eqs.
(1.23)-(1.25) yield:
A(λ) = −iA0
D(λ)
where D(λ) = λ−iκ+∫ +∞−∞ dδ G0(δ)
(λ+δn)2−1/4ρ2). The Fourier transform of the radiation
field is
A(t) = A0F (t)
where
F (t) =∑
Res
(eiλt
D(λ)
)
The growth modes λ are given by the roots of the dispersion equation D(λ) = 0.
Assuming G0(δ) centered around δn = δ + n/ρ and defining ω = λ + δn and
δ′= δ − δn, the dispersion equation becomes:
ω −∆ +
∫ +∞
−∞dδ′
G0(δ′)
(ω + δ′)2 − 14ρ2
= 0 (1.26)
where ∆ = δ + n/ρ − iκ is the generalized detuning. Eq. (1.26) can be written
also in the following form
ω −∆ + ρ
∫ +∞
−∞
dδ′
(ω + δ′)
[G0
(δ′ +
1
2ρ
)−G0
(δ′ − 1
2ρ
)]= 0
Chapter 1. Collective Atomic Recoil Laser 16
In the classical limit ρ À 1 the finite difference ρ[G0
(δ′ + 1
2ρ
)−G0
(δ′ − 1
2ρ
)]
tends to the derivative dG0
dδ′ , so that integrating by parts we have the classical
dispersion relation
ω −∆ +
∫ +∞
−∞
dδ′
(ω + δ′)
(dG0
dδ′
)= 0.
The solution of the dispersion relation (1.26) depends on the initial momentum
distribution G0(δ), in general the exact solution can be obtained only numerically
except for some easy distribution.
Figure 1.4: |Imλ| vs. δ for κ = 0 and different value of 1/ρ: (a) 0, (b) 1, (c) 6,
(d) 10, (e) 14, (f) 20.
When the temperature of the atomic sample is T = 0 K the momentum dis-
tribution is flat and the dispersion relation (1.26) reduces to the following cubic
equation:
(λ− δ − iκ)
(λ2 − 1
4ρ2
)+ 1 = 0 (1.27)
It reduces to its classical expression (1.14) when the average number of photons
scattered per atom is large (ρ À 1). The quantum regime of CARL (QCARL)
occurs (in the good-cavity case κ ≈ 0) when ρ < 1, which means that each atom
Chapter 1. Collective Atomic Recoil Laser 17
scatters less than one photon on average. It is at this low scattering rate that
the quantum nature of the scattering, made by single events of photon kicks,
is manifested. Increasing the number of scattered photons the behaviour become
classical, with a continuous transfer of momentum from the radiation to the atoms.
In the quantum regime the frequency of the amplified scattered field is ω1 =
ω2−ωrec, downshifted by the recoil frequency. In the classical regime this quantum
shift is negligible with respect to the collective recoil shift 2ωrecρ. For ρ ¿ 1, the
unstable solution of (1.27) is approximately λ ≈ 1/2ρ+∆/2− i√
ρ−∆2/4, where
∆ = δ − 1/2ρ, which has a negative part different from zero for |∆| < 2√
ρ.
Hence, the unstable bandwidth in the quantum regime ρ < 1 becomes very narrow
and the atoms scatter quasi-monochromatic radiation. Furthermore, the recoil
shift hinders the atom of reabsorbing the emitted photon acquiring a positive
momentum +2~k and thus the state with m = 1 is much less populated than the
state with m = −1, and the atoms start to populate only the momentum state
with m = −1, moving in the same direction of the pump. It is possible to see that
in this regime the state m = −1 is the only populated state also behind the linear
regime. So the atoms have only two available momentum states, the initial state
m = 0 and the single recoil state m = −1. For this reason the atom dynamics
is described by equivalent Bloch equations for the two momentum states. This
makes the Quantum CARL very similar to a system of excited two-level atoms,
basic ingredient of the usual atomic lasers. However, CARL has the important
difference that the atoms do not decay spontaneously from the initial state to the
lower state, so that the emission is in general coherent. The only decoherence
effect is the decay of the off-diagonal elements of the density matrix, related to the
bunching.
Sofar it has been described only the easiest case when T = 0 K, but it is more
interesting to describe what happens for different initial momentum distributions
and thus different temperatures of the atomic sample.
In the case of a Lorentzian distribution, G0(δ) = (1/π)[σ/(σ2+δ2)], the integral
in Eq. (1.26) can be calculated analytically, giving the following cubic equation:
(ω −∆)
[(ω − iσ)2 − 1
4ρ2
]+ 1 = 0 (1.28)
The gain G = −2Imω can be calculated from Eq. (1.28) both in classical regime
and quantum regime. In the classical case (ρ À 1) the gain has an antisymmetric
shape around δ ≈ 0 and it is approximately proportional to the derivative of the
distribution function G0(δ). On the contrary, in the quantum regime the gain
is symmetric around the quantum resonance δ = 1/2ρ. In fact emission and
Chapter 1. Collective Atomic Recoil Laser 18
absorption rates are well separated (if σ ¿ 1/ρ) so that, around δ = −1/2ρ, the
absorption can be neglected and the gain has a symmetric shape.
In the quantum regime the three roots of the Cubic (1.28) can be evaluated
approximately and the gain G = −2Imω, maximum at resonance (δ = 12ρ
), is
Gmax =√
ρ[√
4 + σ2/ρ− σ/√
ρ]. The full bandwidth of the gain is, then, approx-
imately equal to σ∆ ≈ 2√
4 + σ2/ρ. Hence the gain bandwidth is σ∆ = 4√
ρ for
σ = 0 and increases linearly with σ when σ is much larger than√
ρ, but always
smaller than the quantum limit 1/ρ.
In the next chapters I will describe the setup of the experiment in Tubingen
that first measured the Classical CARL signature.
Chapter 2
Experimental setup
After the theoretical proposal of CARL, experiments have been performed in order
to observe its peculiar features. As we saw in the previous chapter the signature
of CARL is the exponential growth of a seeded probe field oriented reversely to
the pump field. On the other hand atomic bunching and probe gain can also
arise spontaneously from fluctuations with no seed field applied, particularly if
the amplification mechanism is enforced recycling the reverse probe field by a ring
cavity.
The first experimental proof of CARL has been obtained recently [28] in a
system of cold atoms in collision-less environment. In the next chapter I am going
to describe the main feature and results of the experiment running in Tubingen.
In particular I will focus on ring cavity, locking system and cooling, trapping
technologies.
2.1 Ring Cavity
Firstly I will describe the heart of the experiment: the ring cavity. The ring cavity
consists of one plane (IC) and two curved (HR) mirrors, as it is shown in Fig.
(2.2). It has a round trip length of L = 87 mm. Hence the distance between the
resonances of the cavity (free spectral range δfsr) is
δfsr = c/L = 3.4 GHz
where c is the velocity of light. The beam waist in horizontal and vertical direction
at the location of the MOT is respectively wv = 117 µm and wh = 87 µm. This
beam waist corresponds to a cavity mode volume Vmode = π2Lwvwh = 1.36 mm3.
For s-polarization the two curved high reflecting mirrors have a transmission of
Chapter 2. Experimental setup 20
1.5 × 10−6, while the plane input coupler has a transmission of 1 × 10−5. The
finesse of the cavity, F = πδfsr/κC , is reduced to F = 150000 due to depositions
of rubidium on the mirror surfaces. This corresponds to an amplitude decay rate
of κ = πδ/F = 2π × 11 kHz.
Figure 2.1: The graph represent the transmitted signal from the ring cavity. The
width of the Lorentz-distribution which fits our data corresponds to the amplitude
decay rate κc = 2π × 11 kHz.
The whole setup consisting of magnetic coils, wires and ring cavity is placed
inside an ultrahigh vacuum chamber. Heat produced in coils and wires inside the
vacuum is dissipated via a temperature-stabilized cooling rod to a liquid nitrogen
reservoir. A second vacuum chamber, connected to this main chamber, accommo-
dates a two-dimensional magneto-optical trap (2D MOT) which produces a cold
atomic beam directed into the main chamber. Typically 2× 108 atoms are loaded
into the magnetic trap at a temperature of T = 100 µK. The atoms are then mag-
netically transferred into a second and then a third quadrupole trap, whereby the
atoms are compressed adiabatically. The distance of the coils decreases from the
first to the third pair in order to get higher gradients. The magnetic quadrupole
field gradient between the third pair of coils is 160 G/cm in the horizontal and
320 G/cm in the vertical direction. With two pairs of wires separated by 1 mm
and running parallel to the symmetry axis of the coils a Joffe-Pritchard type po-
tential is created [19]. The vertical position of the wire trap can be easily shifted
by the currents in the quadrupole coils. Inside the wire trap the atoms are cooled
by forced evaporation. When the evaporation cooling stage is completed, the cold
atoms are vertically transferred into the mode volume of the ring cavity. One
Chapter 2. Experimental setup 21
Figure 2.2: Scheme of the experimental setup.
of the two counterpropagating modes of the cavity is continuously pumped by a
titanium-sapphire laser. The laser is double-passed through an acousto-optic mod-
ulator which shifts the frequency. The light reflected from the cavity is fed back
via a Pound-Drever-Hall type servo control to phase correcting devices. Because
the cavity mirrors have very different reflectivity for s and p polarization, we can
switch from high to low finesse by just rotating the linear polarization of the in-
jected laser beam. The phase corrections are made via a piezo transducer mounted
to the titanium-sapphire laser cavity and via the acousto-optic modulator. As soon
as the pump mode power builds up in the ring cavity, the collective dynamics re-
sults in light scattering into the cavity probe mode. In the next section we are
going to describe the main features of the experiments briefly described above.
2.2 Magneto-optical trap
Magneto-optical traps are a standard cooling technique. They are very close to
the optical molasses technique, it has just to be added a magnetic field gradient
and optical molasses can be turned into a trap (MOT). The magnetic field is
created by two pair of coils around the atoms, with current in opposite directions,
producing a quadrupole magnetic field, which is zero at the center of the coils and
whose magnitude increases linearly in every direction for small displacements from
Chapter 2. Experimental setup 22
Figure 2.3: Principle of the trapping force in a MOT. The magnetic field removes
the degeneracy of the J = 1 level. The energy of the sublevels Mj depends
linearly on the position because of the quadrupole magnetic field. The two counter-
propagating laser beam are polarized σ+ and σ−, thus the transition is allowed only
toward one of the sublevels. The laser beams frequency is red-shifted respect to
the atomic resonance so the resultant force is toward the MOT center.
the zero point. The magnetic field does not confine atoms by itself, but causes
an unbalance in the scattering forces of the laser beams by producing a variable
Zeeman shift of the atomic hyperfine levels and it is the radiation force which
strongly confines the atoms.
For a J = 0 → J = 1 transition, the constant magnetic field gradient splits
the three sub-level with Mj = 0,±1 of J = 1 with a separation depending on the
atom’s position. The counter-propagating laser beams have circular polarization
and red-shifted frequency with respect to the atomic resonance.
The Zeeman shift causes and unbalance in the radiation force. The frequency
shift caused by the magnetic field can be incorporated in the detuning of the
scattering force, ∆ = ω∓kv− (ω0±βz), so that for two laser beams with opposite
circular polarization,
FMOT = F σ+
scatt(∆0 − kv − βz)− F σ−scatt(∆0 + kv + βz)
≈ −2∂F
∂∆0
(kv + βz) = −αv − αβ
kz, (2.1)
where β = (gµB/~)(dB/dz). The unbalance of the radiation force caused by the
Zeeman effect leads to a restoring force with spring constant (αβ/k). Atoms that
enter the region of intersection of the laser beams are slowed and the position-
dependent force pushes the cold atoms to the trap center, providing an efficient
and easy method to load a large number of cold atoms (up to 1010), to be used in
laser cooling experiments.
Chapter 2. Experimental setup 23
Figure 2.4: Schematic of the MOT. Lasers beams are incident from all six direc-
tions and have helicities (circular polarizations) as shown. Two coils with opposite
currents produce a magnetic field that is zero in the middle and changes linearly
along all three axes.
2.3 Atomic properties of rubidium
We will discuss the atomic properties of Rubidium which are relevant for atomic
cooling and trapping. The optical transitions from the ground state of rubidium
are shown in Fig. (2.5).
The Zeeman energy of the two hyperfine ground states in presence of a static
magnetic field can be expressed by the Breit-Rabi formula for the case of vanishing
orbital angular momentum,
EF,mF(B) = (−1)F 1
2~ωhF
√1 +
4mF
2I + 1x + x2 + const.
x =(gI + gS)µBB
~ωhF
The values of the gyromagnetic factors gI and gS, the nuclear spin I, and the
frequency of the hyperfine transition between the ground state levels ωhF are sum-
marized in Table (2.1). The contribution of the nuclear angular momentum can be
neglected, as g1 ¿ gS. The resulting energies of the nuclear angular momentum
are shown in Figure (2.6).
In typical magneto-static traps the Zeeman splitting gSµB is smaller than the
hyperfine splitting ~ωhF of the ground state, and the equation for the energies can
Chapter 2. Experimental setup 24
Figure 2.5: Hyperfine structure of 87Rb: the D-2 line in the near infrared regime
is commonly used for laser cooling. The atoms can be magnetically trapped in
both the |F = 1〉 and |F = 2〉 hyperfine states of the ground state 5S1/2. The data
were taken from [30].
Figure 2.6: Zeeman-effect of the hyperfine ground state: The energies of the
magnetic sublevels and the parameter (gI + gS)µBB representing the magnetic
field are normalized to the hyperfine splitting EhF of the ground state. It is
possible to magnetically trap the four ’low field seeking’ states |F = 2,mF = ±2〉,|F = 2,mF = ±1〉, |F = 2,mF = ±0〉 and |F = 2,mF = ±− 1〉.
Chapter 2. Experimental setup 25
g-factor nucleus gI 0.995× 10−3
g-factor electron gS 2.0023
nuclear spin I 32
hyperfine energy ωhF 2π · 6.8346826128(5) GHz
Table 2.1: Some atomic properties of 87Rb.
be approximated by
EF,mF(B) = (−1)F
(1
2~ωhF + mF gF µB +
1
16(4−m2)
(gSµBB)2
~ωhF
)+ const.
gF = (−1)F 1
2I + 2gS (2.2)
The first term in Equation (2.2) gives the 6.8 GHz hyperfine splitting of the
Rubidium ground state. The second term describes the linear Zeeman effect. The
third term describes the quadratic dependency of the energy on the magnetic field
strength, the so called quadratic Zeeman effect. The states |F = 2,mF = ±2〉 are
pure sin states, for which the quadratic Zeeman effect is absent.
In the experiment both trapping laser than cooling laser operates at hyperfine
transitions of the D2 line at a wavelength of (780.2nm). The D2 line corresponds to
the transition from the ground state 5S1/2 to the excited state 5P3/2. The hyperfine
structure of these states is shown in Figure (2.5). The MOT and 2D-MOT lasers
drive the transition between the hyperfine states |5S1/2, F = 2〉 to |5P3/2, F′ = 3〉.
The repumper lasers bring the atoms to |5P3/2, F′ = 2〉 from the ground state
|5S1/2, F = 1〉 in such a way that there are not losses of atoms in the magneto
optical trap.
The pump laser is tuned very far, between 0.5 and 10 THz below the Rubidium
D1 line. The atomic sample of 87Rb is prepared in the ground state |F = 2,mF =
2〉.
2.4 Evaporative cooling of trapped atoms
Evaporation is a well-known phenomenon in every day life. It describes the conver-
sion of liquid to gaseous state. In a more abstract sense, evaporation describes the
process of energetic particles leaving a system with a finite binding energy. This
happens naturally since there are always high energy particles in the tail of the
Maxwell-Boltzmann distribution. Since the evaporating particles carry away more
Chapter 2. Experimental setup 26
than their share of thermal energy the temperature of the system decrease. Due
to the lower temperature, the evaporation process slows down, unless evaporation
is forced by modifying the system in such a way that less energetic particles can
escape from the system.
Evaporative cooling cools a cup of coffee and it is employed in technical water
coolers. It has led to the coldest temperature ever observed in the universe (sub-
microkelvin temperatures generated in atom traps). In 1995 evaporative cooling,
with a proper use of laser cooling, turned out to be the key technique to achieve
Bose-Einstein condensation ([1], [7], [12]). Nowadays it is commonly used to con-
densate alkali atoms.
2.4.1 Theoretical model for evaporative cooling
Evaporative cooling is based on the preferential removal of atoms above a certain
truncation energy εt from the trap and subsequent thermalization by elastic col-
lisions. The major requirement for the application of evaporative cooling is that
the thermalization time would be short compared with the lifetime of the sample.
In other words the ratio of good (elastic) to bad (inelastic) collision sets the limit
to evaporative cooling.
Usually the velocities of atoms are well described by a Maxwell-Boltzmann
distribution. The atoms with energy above a certain value εt are eliminated in
such a way that the mean energy of the system is reduced. The system after
the thermalization due to elastic collisions is characterized with a new Maxwell-
Boltzmann distribution with smaller mean value and, hence, less temperature.
The process is shown in Figure (2.7).
The rate of evaporation cooling per atom is given by:
1
τev
=Nev
N∝ σe−η (2.3)
where σ is the elastic cross section and η = εt
kBTis the truncation parameter. As
atoms are evaporated from the trap the mean energy is decreased and hence the
gas cools. As the temperature decreases, η becomes larger and the evaporation
cooling rate is exponentially suppressed. A continuous cooling process could be
realized reducing εt for instance in a way that η remains constant (forced evap-
orative cooling). During cooling also the effective volume is decreasing. Despite
the massive loss of atoms due to the evaporation process, the atomic density in
the trap center can remain constant or even increase. In order to realized such a
density increase large η and thus slow evaporation is necessary. If the increase of
Chapter 2. Experimental setup 27
Figure 2.7: Thermalization of a truncated Maxwell Boltzmann distribution for85Rb atoms (grey line). After the thermalization the velocity distribution of the
atoms is a new Maxwell Boltzmann (dark line) with less temperature
the density is so strong that the elastic collision rate increases, although the tem-
perature decreases, the so called runaway evaporative cooling regime is reached
and the evaporation process proceeds faster and faster for decreasing temperature
[13]. In practice the efficiency of evaporative cooling is limited by atomic loss from
the trap.
The energy selection removal of the atoms from the trap is done by driving
transition to untrapped Zeeman states by applying an oscillating magnetic field
of angular frequency ωrf . As we know the atoms undergo the transitions only at
positions where the resonance condition:
~ωrf = µB|mF gF −mF ′gF ′ |B(−→r ). (2.4)
is fulfilled. Therefore the truncation energy εt at which atoms in the state mF are
removed from the trap is related to the rf frequency ωrf as
εt = mF~(ωrf − ω0)
Chapter 2. Experimental setup 28
where ω0 = µBgF |B(0)|/~ is the resonance frequency at the center of the trap. Due
to their thermal motion the atoms are passing with different velocities through the
trap. This can equivalently be seen as an atom at rest experiencing a time varying
magnetic field. Assuming a linearly polarized oscillatory field with amplitude Brf
along the x-axis perpendicular to the trapping field B(t), the magnetic field can
be expressed as −→B (t) = B(t)ez + Brfcos(ωrf t)ex.
The time dependent Hamiltonian of the system is then
H(t) = ~ω0(t)Fz + ~ΩRcos(ωrf t)Fx
where ~ω0 is the energy of the atoms in the trap, ΩR is the Rabi frequency of the
rf field. In a Landau-Zener picture the transition probability for a two levels atoms
can be calculated
P = 1− e−2πΓ, Γ =Ω2
r
4d∆dt
(2.5)
When ΩR ≥ 4d∆dt
the probability for an adiabatic transition is more than 99.8%.
In order to reach such a probability it is necessary that:
Brf =~γ
ΩR ≥ 2~γ
√d∆
dt= 2
~γ
√dω0
dr
dr
dt= 2
~γ
√γ
~dBz
dr
√v (2.6)
=⇒ Brf ≥ 2
√~γ
dBz
dr
√v
For instance if dBz
dr= 100 G
mat 100µK the transition probability (2.5) is more than
95%.
With Rubidium atoms it is usually driven the transition from | F = 2,m = 2〉to the state | F = 1,m = 1〉. These levels are separated by a frequency of 6.83
GHz, so we use a Microwave Antenna in order to produce the resonant oscillating
magnetic field.
2.4.2 Microwave Antenna
In our experiment it has been used an helix antenna. As you can see in figure (2.8)
the helix is wound to a stick in plexiglass, the support is also made of plexiglass
and the base is made of copper with a SMA-pin. The total length is about 90mm.
We observe also that the impedance of the antenna must be adjusted to avoid
reflection (commonly the resistance is about 50Ω).
The Antenna could be well described by few characteristic quantities [13], in par-
ticular we are interested on the gain, which, for an helix antenna, is a function of
Chapter 2. Experimental setup 29
Figure 2.8: Helix antenna.
the circumference Cλ, the gradient of the helix Sλ and the number of windings n.
The gain in dBi is:
G = 11.8 + 10log(nC2λSλ)
These quantities scale with λ. Our antenna, made of PMMA plexiglass, has a
εr = 2.475 for a frequency of 6.8GHz and at 24 , hence λ is given by:
λ =c0√εrν
=3 · 108 m
s√2.475 · 6.8 · 109 1
s
≈ 28mm
At that wavelength Cλ = 1.2 and Sλ = 0.277, if n = 8 windings the total gain
is of 17.9dBi.
Till now we have described the problem without considering the effect of bound-
ary conditions, in fact inside the chamber we have a very complex environment
made of different materials and surfaces. Because of reflection from the environ-
ment we do observe the formation of standing wave.
It is important to understand if the modulation in the intensity of the mi-
crowave frequency could affect the intensity of the microwave and, hence, the
efficiency of evaporative cooling.
In order to investigate the effect of boundary conditions on the evaporative
cooling process, we realized the experiments shown in figure (2.10). In this ex-
periment we created an artificial boundary condition using an aluminium (and
copper) surface. We shielded also the Microwave Antenna with a pipe covered in
Aluminium foil in order to avoid undesirable reflection from the environment.
Chapter 2. Experimental setup 30
Figure 2.9: Spectrum of the microwave antenna. The maximum emitted fre-
quency is at 6.8 GHz as the evaporative cooling of Rubidium atoms require.
We measured the intensity modulation moving the pick-up coil along a straight
line. The measured standing wave is shown in picture (2.11). As we can see if we
put a reflecting surface we observe a very strong modulation in the intensity of
the wave, while if there are not any artificial boundary conditions the modulation
effect is very weak. Both the signals show an exponential decay in the intensity
due to the increasing distance from the source (Microwave antenna).
The formation of a standing wave could be a problem for the evaporative cooling
process if the atoms sit near a node of the standing microwave. The atoms should
be shined with enough intensity in order to have a reasonably big probability of
evaporation, see (2.5). If that condition is not satisfy it should be necessary to
screen the vacuum chamber with materials that absorb microwave frequencies.
In our experiment the atoms are 3 cm far from the antenna, we verified that
this distant is within the near field range, which is about 5 cm. At that distance
the intensity of the microwave radiation is almost constant.
Chapter 2. Experimental setup 31
Figure 2.10: Scheme of the experiment realized in order to investigate the effect
of boundary conditions on the radiation emitted by the microwave antenna.
Figure 2.11: Modulation of the intensity emitted from a microwave antenna.
When there is a reflecting surface we observe the formation of a standing wave
(red points), when there are not any artificial boundary surface the modulation is
not so strong (green points).
2.5 Two-mode laser locking system
In previous experiments the buildup time for the ring cavity pump mode was
limited by the bandwidth of the locking servo to about 20 µs, which was longer
than the cavity decay time. For the CARL experiment a precise timing of the pump
laser irradiation and the ability to choose the pump laser power and detuning over
Chapter 2. Experimental setup 32
wide ranges are often essential. Hence recently it has been developed a simple and
reliable experimental scheme based on a two-mode laser locking [8]. The optical
layout for the two-mode locking is shown in Figure (2.12).
To satisfy the requirements of tunability, fast switching and variable pump
power it is necessary to reference the laser with respect to the cavity’s mode
structure. The most convenient way is to use the TEM00 mode as pump laser
and an higher transversal mode to probe the cavity’s free spectral range (reference
mode laser).
Figure 2.12: Principle scheme of the optical setup. The Ti:Sa laser is locked via
a PDH servo to a TEM11 mode of the ring cavity. Part of the signal is injected
into a counterpropagating TEM00 mode. The computer control the shutters and
the function generator. This device produces both the microwave frequency for
the evaporative cooling and the modulation frequency which control the AOM.
The Ti:Sa is stabilized to the TEM11 of the ring cavity using the Pound Drever
Hall technique. The fast components of the error signal are fed to an acousto-optic
modulator not shown in the scheme.
We should choose the modes carefully, it is necessary that the reference mode
laser does not interact with the atoms. This kind of interaction results in radiation
Chapter 2. Experimental setup 33
pressure, which heats the atomic cloud and reduces the efficiency of cavity cooling
or perturbs the collective dynamics.
Hence we should avoid radiation pressure by the reference mode laser and, at
the same time, the reference laser power must be sufficiently strong to guarantee
stable operation of the PDH servo. It has been demonstrated [8] that we can
achieve this by tightly locking the Titanium:Sapphire laser to a TEM11 mode. If
the atomic cloud is small enough, it may be contained within a region of space
where the intensity of the TEM11 mode is negligibly small, so that radiation
pressure is efficiently reduced.
Chapter 3
Pound-Drever-Hall laser
frequency stabilization
The Pound-Drever-Hall frequency stabilization is a powerful technique for improv-
ing the frequency stability of a laser by locking it to the mode of a high-finesse
optical cavity [14]. It is often used in atomic physics experiments, metrology and
ultrahigh resolution spectroscopy.
The idea behind the Pound-Drever-Hall method is simple in principle: a laser fre-
quency is measured with a Fabry-Perot cavity, and the signal resulting from this
measurement is fed back to the laser in order to suppress frequency fluctuations.
In this chapter I will describe the main feature of this technique and experimental
results.
3.1 Pound-Drever-Hall frequency stabilization
The best way to measure the detuning of a laser beam is to send it into a Fabry
Perot cavity and observe the transmitted or reflected power. One could control
the laser frequency feeding the transmission signal back to the laser in order to
hold the intensity, and hence the frequency, constant by stabilizing the laser on
the slope of the transmission curve or using a modulation technique. With such
a frequency control system it is possible to suppress fluctuations eliminating any
conversion of laser intensity noise into frequency noise, as it is well explained in
[3].
One possible way to stabilize a laser is to measure the reflected intensity and
hold that at zero (which means that the laser is resonant with the cavity), but the
intensity of the reflected beam is symmetric about resonance, so it is impossible
Chapter 3. Pound-Drever-Hall laser frequency stabilization 35
to know whether the frequency needs to be increased or decreased to bring it back
to resonance. In order to stabilize the laser we should look for a dispersive signal
antisymmetric about resonance.
Figure 3.1: Basic layout for locking a cavity to a laser. A polarizing beamsplitter
(PBS) followed by a quarter-wave plate separates the light reflected from the Fabry-
Perot cavity (FPC) and the incident beam. The reflected beam is detected on a
photodiode. The FM-BOX contains the quarz that generates the modulation signal
at 40 MHz, the mixer (RPD-1) that adds the modulation to the reflected signal
and the attenuator AT-10, AT-5 (indicated with dB in the picture) that regulate
opportunely the amplitude of the error signal. In order to avoid that the reflected
signal goes back to the laser diode generating instability it has been inserted a
faraday isolator (FI). DCC is the DC current driver for the laser diode.
In the picture (3.1) the basic layout for the Pound-Drever Hall technique is
drawn. Via a modulation technique optical sidebands are generated, spectrally
located well outside the resonator pass band. These sidebands are totally re-
flected from the control cavity input mirror, while the laser carrier frequency ap-
proximately matches the cavity resonance and so an intracavity standing wave is
generated at the laser frequency. It is important to note that the leakage field
back toward the source is in antiphase with the input field directly reflected from
the coupling mirror. The approximate cancelation of these two fields (in reflec-
tion) leads to a small net reflection coefficient with a phase shift which is strongly
frequency dependent in the vicinity of the resonance. The phase sensitive demod-
Chapter 3. Pound-Drever-Hall laser frequency stabilization 36
ulation against the local oscillator source converts the symmetric minimum in the
cavity reflection coefficient into the desired antisymmetric frequency discriminator
curve.
A fundamental component of the Pound-Drever-Hall technique is a Fabry-Perot
cavity. We know that a Fabry-Perot cavity consists of two reflectors separated by
a fixed distance L [18]. For two identical mirrors, each with reflectivity R and
transmission T (R +T = 1) without losses, the amplitudes of the transmitted and
reflected electric field are
Et =Teiδ
1−Re2iδEi (3.1)
Er =(1− ei2δ)
√R
1−Re2iδEi (3.2)
where Ei is the amplitude of the incident light and δ = 2πLλ
is the phase shift
of the light after propagating through the cavity. The transmission and reflection
coefficients are then
T =Et
Ei
=Teiδ
1−Re2iδ(3.3)
R =(1− ei2δ)
√R
1−Re2iδ(3.4)
Standard configurations for the Pound-Drever-Hall technique generate side-
bands modulating the beam’s phase by passing the light through an acusto-optical
modulator (AOM). We have been using a simpler setup, in fact we generate the
sidebands modulating in frequency the current of the laser diode, for the complete
circuits see (Appendix A). Actually this is a modulation of the frequency of the
electric field instead of the phase. Anyhow in the following calculation we will
consider the equations for phase modulation because they are easier, but actually
they bring to the same result. The modulation at Ω = 40 MHz is summed to the
diode dc current with a standard two input Bias-T. Hence the modulation current
and phase are
I(t) = I0(1 + β sin Ωt)
φ(t) = φ0 + β sin Ωt
where Ω is the phase modulation frequency and β is the modulation depth. The
modulated laser field can be written as
Einc = E0ei(ωt+βsinΩt) (3.5)
Chapter 3. Pound-Drever-Hall laser frequency stabilization 37
The exponential term in the Eq. (3.5) can be expanded up to the first order using
The Ω terms comes from the interference between the carrier and the sidebands,
while the 2Ω terms arise from the two sidebands interference. We are interested
only in the two terms of Eq. (3.7) that oscillate at the modulation frequency
Ω since they sample the phase of the reflected carrier. Hence in the following
calculations the others terms will be neglected.
−80 −60 −40 −20 0 20 40 60 80−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Frequency (MHz)
Ref
lect
ion
Sig
nal
−80 −60 −40 −20 0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Frequency (MHz)
Tra
nsm
issi
on S
igna
l
Figure 3.2: Pound-Drever-Hall reflection and transmission signal.
The electronic device (mixer) showed in Picture (3.1), mixes the photodiode
output |Er|2 with the demodulation oscillation e−iΩt+iθ, where θ is a generic phase
which takes into account for unequal delays in the two signals paths, thus the
resulting error signal is:
SPDH = |Etot|2e−iΩt+iθ =
= J0(β)J1(β)Reeiθ[E∗r (ω)Er(ω + Ω)− Er(ω)E∗
r (ω − Ω)] (3.8)
Chapter 3. Pound-Drever-Hall laser frequency stabilization 38
The picture (3.2) represent a plot of the Error Signal (3.8) as a function of ω As
expected the reflection signal has an antisymmetric shape and for this reason it is
the core of the Pound-Drever-Hall technique.
3.2 Pound-Drever-Hall setup
As it is shown in the drawing (3.1) a polarizing beamsplitter followed by a quarter-
wave plate separates the reflected light from the incident beam. Afterwards the
reflected beam is detected on a photodiode. The photodiode has to be chosen
carefully, firstly it must be able to acquire the fast signal at 40 MHz and, moreover,
it has to be sensitive enough in order to have, at the end, an acceptable big signal.
The detected signal goes hence through a low-noise amplifier and a high pass filter
Figure 3.3: Pound Drever Hall layout. In the figure the main components of the
setup are highlighted, while the full electronic scheme of our experiment is drawn
in Appendix A.
(see Figure (3.3)) and before reaching the mixer the signal comes through a phase
detector which compensates for different signal paths changing opportunely the
phase. Finally it is mixed to the modulation oscillation and enters the feedback
loop.
Once the error signal (3.2) is generated, it is split off in two parts. The first
part goes through the servo system of amplification and control the piezo of the
Chapter 3. Pound-Drever-Hall laser frequency stabilization 39
laser diode compensating for low frequency noise by changing the length of the
resonator. The second part of the error signal is sent through a Loop-filter to
ensure the feedback is applied with the appropriate phase and, then, it goes to the
current driver of the diode laser. This second feedback loop is able to suppress
fast oscillations.
Thanks to the two feedback loops the Pound-Drever-Hall system is able to
provide corrections for frequency fluctuations over a broad bandwidth, that can
be even larger than 4 MHz. The aim of our experiment was to stabilize a standard
Figure 3.4: Experimental setup for the PDH stabilization technique.
laser diode at 780 nm with the Pound-Drever-Hall stabilization technique. The
experimental setup that we implemented is shown in the Picture (3.3).
The optical cavity that we used is characterized by a free spectral range δFSR =
c/L = 2, 5 GHz and a finesse of F = 600. The Faraday isolator (FI) showed
in Figure (3.1) keeps the reflected beam from getting back into the laser and
destabilizing it.
The main parts of the experimental setup are: the FM-BOX which produces
the modulation signal, the Bias-T, the Lock-Box and the Loop-Filter.
As it can be seen from the electronic scheme (Appendix A), the error signal
comes out of the FM-BOX. In the FM-BOX we find also a quarz, that generates
the 40 MHz signal, the mixer, a phase shifter and three attenuators which allow
Chapter 3. Pound-Drever-Hall laser frequency stabilization 40
to adjust phase and amplitude of the demodulation signal.
Both the Lock-Box and the Loop Filter are on the feedback loop. The Lock-Box
is actually an integrator circuit and it provides a stable overall response function for
the loop. The signal coming out of the Lock-Box goes to the piezo and it stabilizes
the frequency, changing opportunely the length of the cavity’s laser. Typically a
piezo is able to keep the laser locked within few kHz of the cavity resonance.
Figure 3.5: Loop Filter. The loop filter has been improved starting from the
original project of [21].
The Loop-Filter takes care of the fast oscillation. It consists in one attenuator
and two filter circuits (see Fig. (3.5)). It adjusts the amplitude and the phase
of the error signal. In fact, due to the finite time delay in the feedback loop, all
Fourier frequencies of the error signal cannot be sent back to the laser with the
proper phase. The Loop Filter compensates for this delay.
We characterized the Loop Filter measuring its Bode diagrams of amplitude
and phase. Finally the signal, conditioned by the Loop Filter, is sent to the current
driver of the laser. Both the modulation signal at high frequency (40 MHZ) and
the error signal, whose frequency is about kHz, have to be added to the DC current
of the laser. We projected an electronic circuit able to manipulate a broad range
of frequency without damaging the diode. Since the diode is very sensitive to any
parasite DC voltage, we used an appropriate transformer, able to add the high
frequency signal to the DC current, and a RC circuit. Both the transformer and
the RC circuit decouples the DC voltage, so they are appropriate for our purpose.
We chose resistance and capacitor carefully, because we did not want to cut signals
in the range of frequencies from kHz to few MHz.
Chapter 3. Pound-Drever-Hall laser frequency stabilization 41
Figure 3.6: Bode diagrams of amplitude and phase.
Figure 3.7: Bias-T
3.3 Measurements and Data Analysis
In order to understand how good was our Pound-Drever-Hall stabilization we
should compare the situation with and without loop filter. As I explained in
the section before, the feedback through the loop filter suppresses fast oscillations
and, in general, it improves considerably the stability of the laser.
The figure (3.8) shows the error signal and the transmitted intensity, as the
cavity is tuned on the sidebands spectrum of the phase modulated laser. The
upper line indicates the signal obtained without loop filter and the lower down
curve represent the signal measured with loop filter.
We can notice that when the feedback on the current is present, the transmis-
sion signal is broader and the system is trying to lock, even without locking the
Chapter 3. Pound-Drever-Hall laser frequency stabilization 42
20 40 60 80 100 120 1400
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Frequency (MHz)
Vol
tage
(V
)
20 40 60 80 100 120 140−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Frequency (MHz)
Vol
tage
(V
)
20 40 60 80 100 120 1400
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Frequency (MHz)
Vol
tage
(V
)
20 40 60 80 100 120 140
−0.1
−0.05
0
0.05
0.1
0.15
Frequency (MHz)
Vol
tage
(V
)
Figure 3.8: Transmission and Error signal measured
piezo transducer. The error signal shows that the system is trying to bring the
laser back to resonance δω = 0, i.e. where the dispersive signal goes through zero.
Moreover we can observe on the slope of the error signal the servo oscillations,
probably due to an excessive gain of the servo loop. The signals look extremely
noisy probably due to some kind of misalignments in the laser setup. We measured
that the frequency of this noise was more than 3 MHz.
We can perform a more quantitative treatment of our measurements looking
at the Fourier transform of the reflected intensity when the laser is locked.
In the picture (3.9) the upper curves represent the answer of the system without
feedback on the current. We notice that the noise amplitude of the reflected signal
with loop filter is less than in the system without loop filter (lower down curve),
there is almost one decibel of difference.
We realized the Pound-Drever-Hall stabilization for a laser diode at 780 nm.
In particular we developed the electronics of the Bias T in order to make the Loop
Filter working. We optimized the system for a cavity of finesse not very high, so
that the stability improvement due to the loop filter was not evident. Anyhow
we did observe that with the second feedback loop the locking signal was less
noisy and the system less sensitive to external vibrations. We have demonstrated
that the Pound-Drever-Hall is an efficient technique for frequency stabilizing laser
diode. Unfortunately our signals were quite noisy and the finesse of the reference
Chapter 3. Pound-Drever-Hall laser frequency stabilization 43
0 20 40 60 80 100 120 140 160−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
Frequency (MHz)
Vol
tage
(V
)
0 1 2 3 4 5
x 105
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Frequency (Hz)
FF
T (
dBm
)
0 20 40 60 80 100 120 140 160−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
Frequency (MHz)
Vol
tage
(V
)
0 1 2 3 4 5
x 105
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Frequency (Hz)
FF
T (
dBm
)
Figure 3.9: Noise analysis of the data
cavity was still too low. In the future the experimental setup should be improved
for very high finesse cavity eliminating also these high frequency noise that the
PDH cannot suppress.
Chapter 4
Toward the observation of the
quantum regime of CARL
The experimental realization of Bose-Einstein condensates with alkali trapped
atoms has opened the possibility of investigating several fundamental aspects of
quantum mechanics in macroscopic systems. CARL appears as a promising source
of macroscopic entangled or squeezed systems. In particular CARL could create
quantum correlations between atoms with different momentum and between atoms
and radiation. The experiment, presently running in Tubingen, is going in that
direction and the next challenge will be the observation of the quantum regime of
CARL.
As it has been explained in Chapter 1 the quantization of the atomic motion
becomes relevant when each atom scatters less than one photon on average. In a
conservative regime the quantum system depends on a single collective parameter
ρ, which in the classical limit can be interpreted as the average number of photons
scattered per atom. Hence when ρ À 1 the semiclassical regime of CARL is
recovered and many momentum levels are populated. On the contrary when ρ ≤ 1
only two momentum levels are populated, i.e. m = 0 and m = −1.
Experimentally the quantum limit can be observed only for a limited range of
parameters (pump power, detuning, etc....).
In the next chapter we will present the results of a numerical study of CARL
simulated on the computer using parameters close to their experimental values.
The aim is to understand if it is possible to observe the quantum regime of CARL
and find the optimal parameters for a cavity finesse of F = 150000.
Chapter 4. Toward the observation of the quantum regime of CARL 45
4.1 Experimental parameters
In order to derive the CARL Hamiltonian that has been presented in [28] I have
to define the electric field per photon as
ε1 =
√~ω
2ε0VMODE
(4.1)
where ε0 is the vacuum permittivity and VMODE = π2w2 is the mode volume of a
resonator with length L and beam waist w.
We define α+, for the pump laser field, and α−, for the unpumped field, as the
electric field amplitude normalized to the field generated by a single photon:
α± =E±ε1
(4.2)
thus |α±|2 corresponds to the number of photons in the mode. Moreover the dipole
momentum is
d =
√3πε0~Γ
k3
where Γ is the natural linewidth of the D1 line of 87Rb.
The single-photon light shift far from resonance is defined as
U0 =Ω2
1
∆0
(4.3)
where Ω1 = dε1/~ is the single-photon Rabi frequency and ∆0 is the detuning
between the light and atomic resonance frequency. U0 can also be interpreted as
the Rabi frequency for the coupling between the pump and the probe mode, i.e.
the rate at which photons are exchanged between the modes.
Using these definitions the equations of motion for the coupled system can be
written in the form presented in Eqs.(9) of [28]:
dzj
dt=
pj
mdpj
dt= −2i~kU0α+(α∗−e2ikzj − α−e−2ikzj)
dα−dt
= −(κc + i∆c)α− − iU0α+
N∑j=1
e−2ikzj (4.4)
where ∆c is the detuning between pump and probe, κc is the cavity damping. As
we know b = N−1∑N
j=1 e−2ikzj measures the atomic bunching.
Chapter 4. Toward the observation of the quantum regime of CARL 46
Introducing the dimensionless variables θj = 2kzj, t = 2ωrecρt, ωrec = 2~k2/m =
(2π)14.53 kHz and pj = pzj/(2~kρ) the equations become
dθj
dt= pj
dpj
dt= −i
U0α+
2ωrecρ2(α−eiθj − c.c.)
dα−dt
= (−κc + i∆c)α−
2ρωrec
− iU0Nα+
2ωrecρ
1
N
N∑j=1
e−iθj
defining A = iα−(U0α+/2ωrecρ2) = iα−(2ωrecρ/U0α+N) it follows immediately an
explicit form for the parameter ρ and the field A:
ρ =
(U0α+
√N
2ωrec
) 23
A = iα−√ρN
Since the intracavity light intensity for the respective complex mode α± is I± =
2ε0cε21|α±|2 the light intracavity power is
P− =π
2whwvI− = ~ωδfsrρN |A|2 (4.5)
where wv and wh are the vertical and horizontal beam waist, respectively, and
the beam waist id defined as w2 = whwv Finally defining κ = κc/(2ωrecρ) and
δ = ∆c/(2ωrecρ) the equations of motion are reduced to the usual form
dθj
dt= pj (4.6)
dpj
dt= −g(t)(Aeiθj + A∗e−iθj) (4.7)
dA
dt=
g(t)
N
N∑j=1
eiθj + iδA− κ(A− Ain). (4.8)
where the pump profile g(t) has been highlighted and the injected field Ain takes
into account for mirrors backscattering.
In order to introduce the mirrors backscattering we should have added a con-
stant term −iUsα+ in the field equation (4.4). In the scaled form of Eqs. (4.6)-(4.8)
this term becomes +g(t)Us/(U0N) and finally
Ain = g(t)Us
U0Nκ
Chapter 4. Toward the observation of the quantum regime of CARL 47
The quantum limit of CARL is achieved when the gain bandwidth is so small that
only adjacent momentum states are coupled (see Chapter 1). In the good cavity
regime the quantum limit is distinguished by different values of the parameter ρ.
Since ρ can be expressed as
ρ =
(Ω2
1~ωδfsr
4ωrec
)1/3P
1/3+ N1/3
∆2/30
(4.9)
we can observe the CARL signal for different values of ρ changing either the pump
power P+, the detuning ∆0 or the number of condensed atoms N .
Finally we introduce the scaled Doppler broadening:
σ =σv
2ωrecρ
where σv = 2k(kBT/m)1/2 is the Doppler width of the atomic velocity distribution.
4.2 Numerical analysis of CARL’s equations
Figure 4.1: Carl signal in the quantum limit at T = 0 K.
Chapter 4. Toward the observation of the quantum regime of CARL 48
As we will see in the following sections it is necessary to reach temperatures
below 100 nK in order to observe some signature of Quantum CARL. In fact the
temperature strongly affects the atoms-light interaction dynamics.
The typical CARL signal in the quantum limit is represented in Figure (4.1).
It has been obtained simulating the motion’s equations at T = 0 K, P+ = 50 mW
and ∆0 = (2π)1 THz and thus ρ = 0.63 and κ = 0.63.
Even if it is not possible to reach temperatures below the nK with a proper
choice of the experimental parameters it is possible to observe a signal very close
to the signal of Figure (4.1).
4.2.1 Analysis of CARL at different temperatures
Let’s now consider the experimental setup described in the second chapter. In
particular we will simulate, using Fortran 90, the equations of motion for the high
finesse case, F = 150000.
Figure 4.2: Power of the CARL signal versus time. The red curve has been
obtained simulating the classical motion’s equations, while the blue curve corre-
sponds to the quantum equations. The curve are almost identical.
In order to understand when the quantum limit is achieved we should com-
pare the simulations of the quantum motion’s equations to the simulations of the
Chapter 4. Toward the observation of the quantum regime of CARL 49
classical equations.
Firstly we start with temperature T = 1µK, detuning from atomic resonance
∆0 = (2π)1 THz, number of atoms N = 105, pump power P+ = 1 W and a seed
signal amplitude of Us = 0.01. Using the equations illustrated in Section [4.1]
the dimensionless parameters can be immediately calculated starting from the
experimental variables. In particular our values corresponds to ρ = 1.7, σ = 0.5,
κ = 0.23 and Sc = 0.078.
In Figure (4.2) the numerical results are represented. The red line is obtained
simulating the classical CARL equation, while the blue curve is obtained starting
from the quantum CARL equations for the same parameters. We notice that the
two curves almost coincide showing that we are still in the classical regime of
CARL.
In order to achieve the quantum limit we should decrease ρ, for instance de-
creasing the pump power and necessarily also the temperature.
Figure 4.3: Power of the CARL signal versus time. The red curve has been
obtained simulating the classical motion’s equations, while the blue curve corre-
sponds to the quantum equations.
In Figure (4.3) the curves are obtained simulating the classical (red line) and
Chapter 4. Toward the observation of the quantum regime of CARL 50
quantum (blue line) equations at T = 100 nK, P+ = 100mW , that correspond to
ρ = 0.79, σ = 0.34 and κ = 0.5, while the other parameters are unchanged.
In this case the two curves do not coincide, even if the differences are not very
evident. We have not yet a clear signature of the quantum regime of CARL, so we
should further decrease ρ.
If we decrease the pump power till P+ = 50mW at 100 nK the parameter ρ
becomes ρ = 0.63 and σ = 0.43, κ = 0.63. In this case the quantum signature of
CARL becomes more evident, as it is shown in Figure (4.4).
Figure 4.4: Power of the CARL signal versus time. The red curve has been
obtained simulating the classical motion’s equations, while the blue curve corre-
sponds to the quantum equations.
This is actually a very nice example of quantum CARL. If we look at the
momentum distribution (4.5) we see that only two adjacent momentum states are
mainly occupied, the initial momentum state m = 0 and the single-recoil state
m = −1 as it is predicted by the quantum theory of CARL.
The experiment in Tubingen should be able to measure this signal. Notice that
the temperature of 100 nK is, nowadays, a typical temperature for Bose-Einstein
condensates. Furthermore we checked that with these parameters the noise, due
Chapter 4. Toward the observation of the quantum regime of CARL 51
to mirrors backscattering, does not significantly affect the CARL signal.
Figure 4.5: Momentum distribution as function of δ. It is evident that there are
two occupied states: the initial momentum state and the single recoil state, while
the others have a negligible population.
4.2.2 Analysis of CARL for different values of the pump-
cavity detuning
Thanks to the development of the two-laser locking system, described in the second
Chapter, it is now possible to introduce a detuning δ between the pump laser and
the cavity mode. Hypothetically it should be possible to tune the pump laser
between -5kHz and +5kHz far away from the cavity resonance. In this range it
can be measured the gain as a function of the detuning, comparing the CARL
signal for different values of δ.
In Figure (4.6) it is represented the CARL signal as a function of time for
different values of the detuning pump-cavity in the case illustrated in the previous
Chapter 4. Toward the observation of the quantum regime of CARL 52
section, i.e. P+ = 50mW , T = 100 nK, N = 105, ∆0 = (2π)1 THz and thus
ρ = 0.63, σ = 0.43 and κ = 0.63.
We compared the signal for δ = 0 kHz, δ = +(2π)1.8 kHz and δ = −(2π)1.8
kHz. We know that in the classical limit of CARL the gain is maximum at reso-
nance, while in the quantum regime the gain increases with the detuning and it
reaches its maximum value at δ ' ωrec. From the curves in Figure (4.6) we have
another signature of the quantum behaviour of CARL. In fact we observed that
the gain is higher when the detuning is positive δ = +(2π)1.8 kHz. We noticed
Figure 4.6: CARL signal as a function of time for different values of the detuning
pump-cavity.
also that the height of the first peak for δ = +(2π)1.8 kHz is smaller than for δ = 0
kHz and δ = −(2π)1.8 kHz. We still do not completely understand the behaviour
of the first peak power. Investigating the value of the first peak power for different
detunings we observe the strange behaviour illustrated in Figure (4.7).
It has been demonstrated [2] on a statistical mechanics approach that in the
classical regime the saturation power of the CARL signal increases with the de-
tuning, while in the quantum limit it reaches its maximum value for δ ≈ ωrec.
Even looking at the time delay of the first peak, in Figure (4.8), we observe a
Chapter 4. Toward the observation of the quantum regime of CARL 53
Figure 4.7: First peak power versus pump-cavity detuning.
strange behaviour. Above resonance the delay decreases suddenly till a minimum
value of 80 µs.
Figure 4.8: Peak delay versus detuning.
Hypothetically we should be able to distinguish the quantum limit just looking
at the first peak power for different detunings, but we do not have yet explained
Chapter 4. Toward the observation of the quantum regime of CARL 54
the reasons of these differences.
Finally we calculated the CARL gain as a function of δ. The result is repre-
sented in Figure (4.9). As we expect the maximum is above the recoil frequency
ωrec and it decreases for negative values of the detuning.
Figure 4.9: CARL gain as a function of the detuning pump-cavity.
4.3 Conclusions
In this chapter we examined the possibility for the CARL experiment in Tubingen
to observe the quantum limit of collective atomic recoil laser. The simulations
show that at 100 nK with P+ = 50mW , N = 105, ∆0 = (2π)1 THz and hence
ρ = 0.63, σ = 0.43 and κ = 0.63 it is possible to observe a clear signature of the
quantum limit. We should take into account that we could not observe the ’good
cavity’ quantum regime. In fact, in order to reach the quantum limit, it must be
ρ ¿ 1 and the temperature such that σ < ρ [25]. When these conditions are both
satisfied the momentum spread is smaller than the photon recoil momentum ~kand only a single momentum state can be amplified at a time. On the other hand
the ’good cavity’ regime is reached when κc ¿ 2ωrecρ, i.e. the gain bandwidth
overwhelms the cavity decay width. Both the conditions on the cavity losses κ
and the temperature T cannot be satisfied at the same time with a cavity finesse
of F = 150000.
Chapter 4. Toward the observation of the quantum regime of CARL 55
On the other hand the superradiant quantum regime of CARL can be observed
in a relatively small range of parameter, for κc ≥ 2ωrecρ and ρ ¿ √2κ and
temperature such that σ < ρ/κ. In addition we could observe the quantum limit
of CARL from the measurements for different detunings. Both the gain and the
first peak power measurements highlight a different behaviour in the classical and
quantum limit of CARL. In conclusion the CARL experiment in Tubingen should
be able to observe both the classical and quantum regime of CARL.
Chapter 5
Raman CARL in a high finesse
ring cavity
So far we described only the interaction between atoms and far off-resonance op-
tical fields. In this case the dominant atom-photon interaction is two-photon
Rayleigh scattering and the collective atomic recoil lasing causes exponential en-
hancement of the number of scattered photons and atoms. In the classical regime
of CARL the scattered atoms can experience further collective scattering, leading
to the observed superradiant cascade.
Recently two experiments [27]-[31] have demonstrated the Superradiant Raman
scattering from a 87Rb condensate, in which the atoms remain, after the process,
in a different hyperfine state not resonant with the pump laser beam, so no fur-
ther scattering of pump photons occurs. As we will describe in the chapter, this
phenomena is well described by a quantum theory.
A Raman CARL experiment allows to observe the quantum limit of CARL
within a larger range of parameters. In the following chapter we will discuss the
general theory of Raman CARL and we will investigate the possibility of observing
it in the high-finesse ring cavity developed in Tubingen.
5.1 Quantum theory for Raman collective atomic
recoil laser
We consider a cloud of BEC atoms which have three internal states labeled by
|b〉, |c〉 and |e〉 with energies Eb < Ec < Ee. The interaction scheme is shown in
Figure (5.1). The two lower states |b〉 and |c〉 are coupled to the upper state |e〉via, respectively, a classical pump field and a quantized probe field of frequencies
Chapter 5. Raman CARL in a high finesse ring cavity 57
ω2 and ω1 in the Λ-configuration.
Figure 5.1: Three level atoms coupled to a quantized probe laser a1 and a classical
coupling laser Ω of frequency ω1 and ω2, respectively. |b〉 and |c〉 are two hyperfine
levels of the ground state.
The second quantized Hamiltonian to describe the system at zero temperature
is given by
H = Hatom + Hatom−field (5.1)
where Hatom gives the free evolution of the atomic fields and Hatom−field describes
the dipole interaction between the atomic field and the pump, probe fields. We
assume the condensate to be sufficiently diluited in order to neglect the nonlinear
atom-atom interaction. Within this approximation, the condensate is described
by a single-particle Hamiltonian of N atoms in a self consistent optical potential.
The Hamiltonian is given by
Hatom =∑
a=b,c,e
∫d3xψ†α(−→x , t)
[− ~
2
2m∇2
]ψα(−→x , t) (5.2)
where ψα(−→x , t) and ψ†α(−→x , t) are the boson annihilation and creation operators
in the interaction picture for the |α〉-state atoms at position −→x . They satisfy the
standard boson commutation relation [ψα(−→x , t), ψ†β(−→x′ , t)] = δα,βδ(−→x − −→x ′) and
[ψα(−→x , t), ψβ(−→x′ , t)] = [ψ†α(−→x , t), ψ†β(
−→x′ , t)] = 0.
Chapter 5. Raman CARL in a high finesse ring cavity 58
The atom-laser interaction Hamiltonian is
Hatom−field = −~∫
d3x[1
2Ωψ†e(
−→x , t)ψb(−→x , t)ei(
−→k 2·−→x−∆2t)+
+g1a1(t)ψ†e(−→x , t)ψc(
−→x , t)ei(−→k 1·−→x−∆1t) + H.c.]
(5.3)
where ωb,c = (Ee −Eb,c)/~ are the resonant frequencies for the two atomic transi-
tions, ∆2 = ω2−ωb, ∆1 = ω1−ωc, g1 = µceε1/~ and Ω = µbeE2~ with µαβ denoting
a transition dipole-matrix element between states |α〉 and |β〉, ε1 =√~ω1/2ε0V is
the electric field per photon for the quantized probe field of frequency ω1 in a mode
volume V , and E2 is the amplitude electric field for the classical pump laser beam
of frequency ω2. Finally a†1 and a1 are photon creation and annihilation operators
for the probe field.
Let’s consider the case where the pump laser is detuned far enough away from
the atomic resonance that the excited state population remains negligible. In this
regime the atomic polarization adiabatically follows the ground state population,
allowing the formal elimination of the excited state atomic field operator.
Writing the Heisenberg equation for ψe exp[i(−→k 2 · −→x −∆2t)] in the adiabatic
approximation we obtain an expression for ψe(−→x , t).
Finally substituting ψe(−→x , t) into the Hamiltonian (5.3) we arrive at the fol-
lowing effective Hamiltonian:
H =∑
a=b,c
∫d3xψ†α(−→x , t)
+i~g∫
d3x[a†ψb(−→x , t)ψ†c(
−→x , t)eiθ −H.c.]− ~δa†a (5.4)
where g = g1Ω/2∆2, a = ia1eiδt, θ = (
−→k 2 − −→
k 1) · −→x and δ = ∆2 − ∆1 =
ω2 − ω1 −∆cb, with ∆cb = (Ec − Eb)/~.As it is described in [11] we can perform the expansion on momentum eigen-
states [23]:
ψb = C
+∞∑n=−∞
bneinθ ψc = C
+∞∑n=−∞
cneinθ (5.5)
where [cn, c†m] = δn,m, [bn, b†m] = δn,m,[bn, c†m] = [bn, c
