IOP PUBLISHING

JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

doi:10.1088/0953-4075/43/1/015301

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 015301 (8pp)

Enhanced Pauli blocking of lightscattering in a trapped Fermi gas
B Shuve1 and J H Thywissen
Department of Physics and Centre for Quantum Information and Quantum Control,
University of Toronto, 60 St George, Toronto, Ontario M5S1A7, Canada
E-mail: joseph.thywissen@utoronto.ca

Received 26 October 2009, in final form 22 November 2009
Published 16 December 2009
Online at stacks.iop.org/JPhysB/43/015301
Abstract
Pauli blocking of spontaneous emission by a single excited-state atom has been predicted to bedramatic at low temperature when the Fermi energy EF exceeds the recoil energy ER . The
photon scattering rate of a ground-state Fermi gas can also be suppressed by occupation of the
final states accessible to a recoiling atom; however, suppression is diminished by scattering
events near the Fermi edge. We analyse two new approaches to improve the visibility of Pauliblocking in a trapped Fermi gas. Focusing the incident light to excite preferentially thehigh-density region of the cloud can increase the blocking signature by 14%, and is mosteffective at intermediate temperature. Spontaneous Raman scattering between imbalanced
internal states can be strongly suppressed at low temperature, and is completely blocked for a
final state EF > 4ER in the high imbalance limit.
In this work, we review two scenarios in which Pauliblocking has been considered previously, and then consider
two new scenarios in which the experimental signature ofblocking can be enhanced. Whereas pioneering work treateduntrapped gases [3, 20, 22, 25], or geometries with spherical[4, 23, 24] or cylindrical [4, 5, 28] symmetries, we treat ageneralized scenario that includes finite temperature and a triaxial trap potential. Since the high optical density of a trappedgas requires off-resonant excitation to avoid the multiplescattering regime, we focus on scattering suppression [4, 5,
28, 7] instead of line shape [3, 6, 20, 27].
We develop, in sections 2 and 3, a semiclassical approachthat may be applied to situations in which fully quantum
calculations have proved onerous. In section 3 we showthat our approach reproduces fully quantum calculationsin the literature. We then calculate angle-averaged finitetemperature signatures without imposing any symmetry.
Finding that suppression is rarely complete, we evaluate
in section 4 two approaches to stronger blocking: using a
focused excitation beam, and scattering between imbalanced
populations. Finally, experimental prospects are discussed in
section 5.

1. Introduction
The Purcell effect is the enhancement or reduction of scattering
due to a modification of the electromagnetic density of statesby a cavity [1]. A complementary effect can occur in an
ensemble of fermions, where final states of the recoiling
particle are blocked, reducing the scattering cross section.
In semiconductors, such blocking creates a Moss–Burnstein
shift [2] of the apparent band gap. Further study of thisfundamental effect has been proposed for neutral Fermi gases[3–7], motivated by the control of trap dimensionality, thedirect quantification of density and the absence of additional
scattering phenomena.
Light scattering is the primary tool for detection ofultra-cold atoms [8], with a few notable exceptions [9]. In
addition to direct imaging, light scattering has been used toprobe density [10], phase coherence [11, 12], momentum
distributions [13], excitation spectra [14] and superradiance[15–17] in quantum degenerate Bose gases. The lightscattering properties of degenerate Fermi gases, by contrast,
have only recently been studied experimentally [18, 19],
despite numerous theoretical proposals [3–7, 20–29]. In situ
optical probes could be particularly useful in exploring thephysics of paired superfluids [20–22, 24, 29]. In addition, thetemperature dependence of light scattering could be exploitedfor thermometry.

2. Methods
We consider light scattering and spontaneous emission in thepresence of a Fermi sea of neutral atoms. Atomic excitation is1

Present address: Lyman Laboratories, Harvard University, Cambridge MA
02138, USA.
0953-4075/10/015301+08$30.00

1© 2010 IOP Publishing Ltd Printed in the UK

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 015301

B Shuve and J H Thywissen

assumed to be far below saturation, and multiple scattering isassumed to be weak. The N degenerate fermions are trappedin a three-dimensional harmonic trap with trap frequencies{ω1 , ω2 , ω3 } along three axes. When a subscript is not
specified, ω refers to the geometric mean of frequencies. The
single-particle Hamiltonian is 11p̂j2 + mωj2 q̂j2 ,
(1)
Ĥ =
2m
2
j

quantum statistical occupation function, the relative scattering
rate is 3 3h−3d p d q ni (p, q ){1 − nf (p + k, q )}. (4)S(k) =
d3 p d3 q ni (p, q )h−3This local density approach neglects the energy quantization
scale set by the level spacing, and thereby allows us torescale the problem into an isotropic one, even though no such
symmetry is manifest in the trap geometry. The symmetry is
broken only by the direction of the momentum recoil due toscattering4 . A semiclassical approximation was also used in[27] to discuss line shape, and to treat the uniform gas in [25].
Here we extend these treatments to find the light scattering
properties of a trapped gas.
For large atom numbers and moderate trap anisotropies,
we show (in section 3) that (4) is in excellent agreementwith published calculations based on (2). Furthermore, thesimplicity of the semiclassical method allows us to include
angular averaging across the scattered photon momentum, toevaluate both finite and zero temperatures and (in section 4) toconsider more complex scenarios.

where m is the mass of the atom, p̂j is the momentum operator
in the j th direction and q̂j is the position operator in the j th
direction.
We treat spontaneous emission and scattering of anincident photon using a golden rule approach, as in [4, 5]2 .
The reduction of the scattering rate by Pauli blocking is
proportional to the reduction in the number of final states,
weighted by matrix elements. This is conceptually similar tothe suppression of spontaneous emission in an optical cavity
where the density of electromagnetic states is reduced; here,
the density of available atomic states is reduced. We define therelative scattering rate S to be the ratio of the scattering ratewith fermions to the scattering rate with Boltzmann particles,
i.e. the rate without blocking effects:
 
ik·q̂
|v |2u ni (v ){1 − nf (u)}|u|e,(2)
S(k) = v 
i
k
·
q̂
2
|v |
v
u ni (v )|u|e

3. Signatures of Pauli blocking
Two scenarios for Pauli blocking have been considered in
the previous work. The first is spontaneous emission (SE)of a single excited-state atom in the midst of a Fermi sea.
The second is light scattering (LS) off a large ground-state
ensemble. For reasons discussed further in section 3.1 and insection 5, the LS scenario is experimentally more feasible, andis the focus of our work. However, the SE case is the simplest
scenario that elucidates the blocking effect under discussion.
Our treatment can also be compared directly to [4], which
presents a fully quantum calculation of the same scenario.

where k is the recoil momentum of the atom, |v  and |u areenergy eigenstates |v1 , v2 , v3  and |u1 , u2 , u3  and ni and nf
are the initial and final occupation functions, respectively3 .
The matrix element along a single direction can be calculatedusing [32]w!
2L [(kj x0j )2 ]2 ,(3)|uj |eikj x̂j |vj |2 = e−(kj x0 )(w + )! w
where kj is the projection of k along the j th direction,
x0j = h̄/2Mωj is the ground-state width, w = min(uj , vj ), = |uj − vj | and Lαn (z) is the generalized Laguerre
polynomial.
Due to the computational demands of a six-dimensionalsum, (2) is more easily calculated in the case of a sphericallysymmetric trap, where three of the sums can be eliminated [4].
A similar approach applied to the case of cylindrical symmetry
can reduce six sums to five. Without these symmetries, ourdesktop computer did not have the resources to calculate S
with experimentally realistic parameters (e.g. N = 106 atoms,
and trap frequencies of ω/2π

= {500, 800, 30} Hz) at finite
temperature. Indeed, no such calculation has been published.
If many states are occupied along all three harmonic axes,
a semiclassical integral may capture the important physics ofthe problem. One might also expect a local density approach
to be appropriate since light scattering predominantly probes
high-momentum properties that depend upon local density
fluctuations in the gas. Starting from the semiclassical phasespace element [33] dN = h−3 n() d3 p d3 q , where n() is the

3.1. Spontaneous emission of a single excited atom
For a single atom in the excited state decaying into a Fermi
sea of N atoms,ni () = (βh̄ω)3 exp(−β)
nf () = [1 + z−1−1

exp β] ,

and
(5)

−1

where β = kB T , kB is the Boltzmann constant, T is thetemperature and z is the fugacity of the Fermi gas. This ni isnormalized by integration over h−3 d3 p d3 q .
Note that we assume that the excited-state atom sees thesame trapping potential as the ground state, and is thermalizedwith the atoms in the ground state. This might be realistic forfermions with long-lived metastable states, for instance in rareearth metals [34] in magic wavelength traps [35].
Using distributions (5), we evaluate (4) using a changeof variables. The coordinates are rotated such that one
momentum axis is aligned with the momentum kick k, and theother five coordinates are combined into a five-spaceradius.
√We define a dimensionless momentum ξ ≡ βER from the2 This approach does not treat coherent effects, which are expected to beimportant within the forward diffraction. cone [8, 23].
3 In Fermi gases, unlike in Bose gases, neglecting the zero point energyh̄ j ωj /2 when calculating the occupation ni,f leads to a fractional error in
the chemical potential of (1 + /2)(6N )−1/3 , where eccentricity  ≡ ωz /ω⊥ .

4 For instance, illumination of a cigar-shaped cloud along one of its two radial
axes would break all rotational symmetries.

2

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 015301

B Shuve and J H Thywissen

Figure 2. Spontaneous emission rate SSE versus momentum kick
κ = (ER /EF )1/2 . Solid lines are numerical integrals as defined in(6), for kB T /EF = 0.5, 0.2 and 0.1, as labelled. Short-dashed lines
are the constant kick approximate form of (10); the long-dashed lineis the zero-temperature limit (8).

Figure 1. Comparison of quantum (points) and semiclassical (lines)
calculation of spontaneous emission blocking. Suppression isabsent when S → 1 and complete when S = 0. For comparison to
figure 1 in [4], the quantum calculations assume ER = 25h̄ω andspherical symmetry; however, these assumptions are not necessary
for the semiclassical calculation. From top to bottom, EF /ER = 0.4,
0.8, 1.2, 1.6, 2.0. The inset ‘cartoon’ is a schematic representationof the process under consideration: an atom in the midst of a Fermi
sea emitting a single photon.

lowest observed temperatures of approximately 0.03EF /kB[36, 37], current experiments remain in the semiclassicalregime.
Figure 2 shows that the emission rate SSE (κ, T 
0.5TF )  0.86, confirming that blocking is not dramatic in
the non-degenerate regime. However, for the weak effects
observed at high temperature, one can use a series expansionof f5/2 :single-photon recoil energy ER = h̄2 kL2 2m, where ckL is thelaser frequency. The normalized scattering rate is then
 ∞
1
2
2
dy e2ξy f5/2 (z e−(y+ξ ) ), (6)SSE (ξ, T ) = √ z−1 eξπ−∞where fn is the Fermi function5 . Note that since we are using a
local density method, trap frequencies and atom number now
affect the scattering rate only through EF .
Figures 1 and 2 show the essence of the blocking effect: at
low temperature, and when the Fermi energy is larger than therecoil energy, the scattering rate decreases. Figure 1 compares
our results to Busch et al [4] to find that the semiclassical
SSE agrees well with a fully quantum calculation, in this casecalculated using (2) and assuming spherical symmetry.
Figure 2 shows that complete suppression of spontaneousemission can occur when the Fermi energy exceeds the recoil
energy. (By contrast, suppression is only complete for infinite
Fermi energy when the set of initial states is expanded to afull Fermi sea, as discussed in the next section.) At zerotemperature, (6) takes the simple form
SSE (κ) =(κ − 1),

SSE (ξ, T ) −→ 1 − e−ξ

2∞(−1)n zn
n=2

n3

2

eξ /n .

(9)This expression converges for z < 1, i.e. kB T /EF > 0.57.
An approximate form valid for all temperatures can be
developed by neglecting the directionality of the momentum
kick. Given an initial atomic momentum p, the average energy
transferred by a kick is simply ER when averaged over a
uniform distribution of atomic momenta. Using this energy
difference, we can fully integrate S:
SSE (ξ, T ) ≈ z−1 eξ f3 (z e−ξ ).
2

2

(10)

Figure 2 compares the approximations (8) and (10) tonumerical integration of SSE . The average-kick approximation(10) underestimates the blurring of the step function at finite
temperatures, but is a reasonable estimate at the 20% level andeven better at low κ. In all cases, suppression is stronger for
lower recoil momentum (or higher Fermi energies), since final
states fall closer to the centre of the Fermi sea.

(8)

where is the unit step function, and we use a dimensionless
momentum κ ≡ (ER /EF )1/2 since ξ is ill-defined atzero temperature.
The abruptness of this step is the
only discrepancy between the quantum and semiclassicalcalculation, as shown at the left-most points in figure 1.
Interesting non-isotropic effects have been predicted in thislimit [4], with possible application to directional single-photon
sources [7]. Wavefunction features appear when kB T iscomparable to the level spacing h̄ωj . However, even at the3.2. Light scattering from a large ensemble
We now consider N polarized fermions in a single groundstate, recoiling under the net momentum k of an incident anda scattered photon. In the perturbative limit, we ignore thedisturbance of removing an atom from the distribution, anduse the same initial and final state:ni () = nf () = [1 + z−1 exp β]−1 .

5The Fermi integrals that appear in this section are of the form
 ∞1
= (n)fn (C),a n−1 da a(7)e /C + 10j nwhere fn (C) is −Lin (−C), and Lin (C) = ∞j =1 C /j is the polylogarithmic
function.

(11)

The initial states have an energetic range that is determinedboth by temperature and Fermi pressure, unlike the case of asingle excited-state atom. This makes it easier to scatter outof the Fermi sea, and reduces the net blocking effect.
3

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 015301

B Shuve and J H Thywissen

Figure 4. Angle-averaged scattering rate MLS versus average kick κ
for a dipole emission pattern. Thick lines are numerical integrals,
for kB T /EF = 0.6, 0.2 and 0.05. The dashed line is thezero-temperature limit. Inset: scattering rate SLS versus momentum
kick κ(α), for the same temperatures.

Figure 3. Scattering rate SLS versus momentum kick κ(α)
comparing fully quantum (points) and semiclassical (lines)
calculations, from equations (2) and (12) respectively. Temperatures
shown are kB T /EF = 1.0 (triangles), 0.47 (circles), 0.24 (squares)
and 0.10 (inverted triangles). The fully quantum calculation is for acylindrically symmetric geometry (ω1 = ω2 ≡ ω⊥ , and incident
beam along axis of symmetry), and parameters are chosen to
reproduce figure 3 in [5]: ER = 21h̄ω⊥ and aspect ratio is 0.1.
Assumptions of symmetry, aspect ratio and ER /ω⊥ are not used forthe semiclassical calculation.

As before, we can expand S to find a high-temperature
expression:∞

SLS (ξ(α), T ) −→ 1 −Integration of (4) using (11) yields a normalized scattering
rate:

 ∞
8
SLS (ξ(α), T ) = 1 −a 3/2 daπ(βEF )3 0
 ∞
1
1
dy
,
(12)
×
2
−1
a+y
−1
1+z e
1 + z ea+(y+ξ(α))2−∞where the results now depend on the angle α between theincident and scattered photon. The rescaled recoil momenta
are ξ(α) = 2ξ sin2 (α/2) and κ(α) = 2κ sin2 (α/2), and we
maintain the previously defined quantities ξ 2 = βER and κ 2 =ER /EF , written without an angle argument. Figure 3 shows anumerical integration of (12) at various temperatures. Points
show a reproduction of quantum mechanical calculations
assuming cylindrical symmetry, for values chosen in [5]. The
agreement is excellent. Unlike the SE case, there is no
disagreement between quantum and semiclassical calculations
at low temperature so long as EFh̄ωj in all directions.
Since the excitation and decay both contribute a
momentum kick, an angle-resolved experiment would observe
S directly [5]. If an experiment measures the total scattering
rate (see discussion in section 5), we observe an angleaveraged suppression factor, which we call M: π αM(k) =
P (α) dα,
(13)
S 2k cos2
2
0

×

n−1


 (−1)n zn
6(βEF )3 n=2n3e−ξ(α) (n− )/n ;
2

(14)

=1

however, the series converges only for z > 1, where
SLS > 0.91. Keeping only the first term, SLS (ξ(α)) ≈1 − 34 (βEF )−3 z2 exp(−ξ(α)2 /2).
At zero temperature, an analytic expression for SLS canbe found:κ(α)
32χ(2 − κ(α)),(15)
SLS (κ(α)) = 1 −
5π
2
where

√
x 1 − x2
15
χ (x) ≡(−8x 4 + 26x 2 − 33) +cos−1 (x).
48
48
(16)
A further integral can also be done to find an expression for
MLS at zero temperature, and is given in the appendix. Bothscattering rates are plotted as dashed lines in figure 4, andlinearly approach zero as the momentum kick goes to zero.
Quantum corrections to the scattering rate can also be
evaluated by considering (2) for various trap geometries,
and comparing to the geometric insensitivity valid in theh̄ωj for all i = {1, 2, 3} trap
semiclassical limit of EF
frequencies. Evaluating (2) for a cylindrically symmetrictrap with 20:1 aspect ratio, and comparing to a sphericallysymmetric trap, both for EF = 11h̄ω, we find that SSE changes
by less than 5%, SLS changes by less than 3%, and MLS changes
by less than 2%.

where P (α) = 38 (1 + cos2 α) for a dipole emission pattern of
any polarization, after averaging over the azimuthal scattering
angle. Numerical evaluation of MLS versus κ, no longer αdependent, is shown in figure 4. Comparing MLS and SLS ,shown in the inset, we see that angle averaging produces
little qualitative change. The low-κ limit is identical, but
suppression continues to higher κ in MLS . This is due
to inclusion of forward-scattering events that produce small
kicks and are easy to block. Including these scattering
events is necessary for quantitative prediction of the scattering
suppression.

4. Scenarios with stronger Pauli blocking
While spontaneous emission can be suppressed fully at finite
momentum κ for sufficiently low temperature, complete
blocking is possible only at κ = 0 in the case of light scattering
4J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 015301

B Shuve and J H Thywissen(section 3.2). The smallest κ reported to date is ≈ 0.6[30, 31, 38], so a suppression of at most MLS ≈ 0.5 would be
expected. In this section, we explore two methods to improve
the Pauli blocking signal: the use of a focused excitation beam(‘FB’), and scattering between two imbalanced populations(‘IP’). In both cases, we attempt to bias scattering towards
events with higher local EF . In the FB scheme, this is done
directly by selecting the spatial centre of the trap. In the
IP scheme, reduced Fermi pressure in the initial state also
reduces the initial kinetic energy. As is shown below, theseapproaches increase the overlap between accessible final states
and the Fermi sea of occupation.
Figure 5. Pauli blocking is enhanced when the excitation beam is
focused onto the centre of the cloud. The normalized scattering rate
SFB is shown versus beam size σ/RF , for κ(α) = 0.5 andkB T = 0.2EF . The arrows indicate the asymptotic limits: the2D
three-dimensional SLS at high σ , and the two-dimensional SLSat
low σ . Inset: the same quantities are plotted with a dipole patternangle-averaged scattering rate MFB .

4.1. Focused excitation light
DeMarco and Jin suggest that stronger blocking might beobserved when focusing the incident laser beam [5]. Here weevaluate this scheme quantitatively. We consider excitation
along a cycling transition, starting and ending in the same
Fermi sea, as in section 3.2. A focused excitation beam
restricts ni to its intersection with the atomic cloud. Assumingthat the beam propagates along q3 , the distribution of initialstates isni (, q1 , q2 ) = I (q1 , q2 )[1 + z−1 exp β]−1 ,would be observed for a two-dimensional cloud with a uniform
excitation light. This limit is
 ∞
4
2D
SLS
(ξ(α), T ) = 1 −a 1/2 daπ(βEF )2 0
 ∞
1
1
dy
.
(20)
×
2
−1
a+y
−1
1+z e
1 + z ea+(y+ξ(α))2−∞Figure 5 shows as arrows the 3D limit (12) and the 2D limit(20).
2D
Figure 6 shows the difference S ≡ SLS− SLS (and2D
angle-averaged M ≡ MLS − MLS ) between the small- andlarge-beam limit. This is the maximum effect that changing
beam size could have. We see that the difference is restricted
to S  0.15. Interestingly, the most pronounced effectoccurs when kB T /EF ≈ 0.25. At intermediate temperatures,
selecting the centre of the cloud is even more important thanat zero temperature, since quantum degeneracy varies acrossthe cloud. At lower temperatures and momenta, suppression
is complete for both the 2D and 3D limits, so focusing is lesseffective.
At zero temperature, an analytic expression can be found:(2 − κ(α))2DSLS[κ(α) 4 − κ(α)2 (κ(α)2 − 10)(κ(α)) =12π−1+ 24 cos (κ(α)/2)].
(21)

(17)

where I (q1 , q2 ) is a dimensionless intensity distribution of thelight. The final-state distribution nf remains as before, givenby (11).
For simplicity we consider
a cylindricallysymmetricbeam I (q1 , q2 ) = exp −2q12 σ 2 − 2q22 σ 2 , where σ is thewaist of the beam. Starting from (4), we rescale symmetricdegrees of freedom and are left with a triple integral:
 ∞
 ∞
2
1
SFB (κ(α), T , σ ) = 1 −a 2 dax dx
π Ne 002
1e−2(xRF /σ )× dy
2 +x 2 )2 +x 2 ) ,
−1
βE
(a+y
−1
βE
(a+(y+κ(α))
F
F
1+z e
1+z e
(18)
where the mean number of atoms excited by the probe is
 ∞
 ∞
2
2
Ne =
a da
x dx e−2(xRF /σ ) [1 + z−1 eβEF (a+x ) ]−1 ,
0

0

(19)
√
the radial Fermi radius RF = 2EF /m/ω⊥ and we haveassumed ω⊥ ≡ ω1 = ω2 . Figure 5 shows that smaller
beam size enhances the suppression. Atoms are excited at
the centre of the cloud, where the density is higher and thus
the local EF is higher. Since ER is unchanged, we effectively
decrease κ.
In the large cloud (or small beam) limit, σRF ,the spatial selection of the exciting beam becomes a delta
function. Since rescaled quadratic degrees of freedom areequivalent under the integral, eliminating two spatial degrees
of freedom is equivalent to eliminating one spatial and one
momentum degree of freedom. In other words, for a given
geometric mean ω, the same scattering rate is observed for atightly focused beam on an oblate three-dimensional cloud, asThis expression is shown as a dashed line in figure 6.
A similar expression can be found for the angle-averaged
2D
MLS, and is given in the appendix. For a variety oftemperatures, figures 5 and 6 show the angle-averaged results
as insets. In the inset of figure 6, the dashed line shows
the zero-temperature difference between equations (A.1) and(A.2). As with angle-resolved scattering, the enhancementM is no more than 0.15, and occurs at intermediate
temperature. However, because of the inclusion of low-α
events, suppression is observed (and enhanced) for κ  2.
4.2. Imbalanced Fermi gases
An alternate method of reducing the distribution of initial states
is to use the internal structure of the atoms. Consider Raman
5

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 015301

B Shuve and J H Thywissen

Figure 7. Scattering rate SIP versus imbalance δ, at κ(α) = 0.5.
From top to bottom, solid lines represent temperatures kB T /EF = 1,
0.5, 0.3, 0.2 and 0.1 from (23); the dashed line shows thezero-temperature limit from (24). Note that for figures 7 and 8, thefinal-state√ Fermi energy has been used for dimensionless quantities,so κ ≡ ER /EFf .

Figure 6. The maximum possible strengthening of Pauli blocking
2D− SLS , is shown versus momentum
with a focused beam, S = SLS
κ(α) for temperatures kB T /EF = 0.5 (cyan), 0.4 (green), 0.3 (red),
0.2 (purple), 0.1 (blue) and zero (black dashed). Inset: the2D− MLS versus normalized recoil
enhancement M = MLS
momentum κ, after averaging over a dipole angular distribution. For
clarity, fewer temperatures are shown: kB T /EF = 0.5 (top solid,cyan), 0.2 (bottom solid, purple), 0.1 (middle solid, blue) and zero
(black dashed). Again the largest effect is at intermediate
temperatures.
(This figure is in colour only in the electronic version)light scattering between two ground states, and the populationof atoms split unequally between them [36, 37] such thatzi < zf , where i and f indicate the initial and final states.
Now the thermalized initial and final distributions are−1ni () = 1 + zi−1 exp β ,
and


−1
nf () = 1 + zf−1 exp β .
(22)Figure 8. Zero-temperature scattering rate SIP versus kick κ(α), forvarious imbalances: δ = 0, 0.5, and 0.99. This corresponds tobalanced, imbalanced and nearly polarized. Inset: angle-averaged
scattering rate MIP versus kick κ and at the same three imbalances.
Imbalance allows complete suppression to be observed fornonzero κ.

As before, we ignore the change in either distribution due toscattered light or due to interactions. We also assume thatincident light excites only atoms from ni and decays only to
nf . 6 Integration of (4) with (22) yields
 ∞8
SIP (ξ(α), T , δ) = 1 −a 3/2 daπ(βEFf )3 0
 ∞
1
1
×
dy
,
(23)
−1 a+y 2−1 a+(y+ξ(α))2
1 + zi e
1 + zf e−∞√
where b = 1 − δ, B = (δ − κ(α)2 )/(2κ(α)) and χ isdefined in (16). Figure 8 shows zero-temperature suppression
at various imbalances. In the limit δ = 0, there is no imbalanceand (24) becomes (15). The comparison in figure 8 makes
especially clear the wide range of suppression possible atfinite recoil momentum for imbalanced gases. Complete
suppression is possible even in the angle-averaged case forκ < 0.5, i.e. for EF > 4ER . By comparison, in the balanced(or single ground-state) case SLS ≈ 0.48 for κ(α) = 0.50.
In the strong polarization limit δ → 1, (24) becomes(κ(α) − 1). This is reminiscent of the spontaneous emission
case (8), apart from the angular dependence of κ(α). We can
now see new significance in the results of section 3.1: in thelimit of extreme polarization, the Raman scattering problem is
equivalent to the spontaneous emission problem with a twophoton recoil momentum. In both cases, suppression is strong
because a second state allows initial states to be exclusively at
the middle of the Fermi sea.
Comparing the two enhancement scenarios, the FB
approach is less effective than the IP scheme, since initialstate selection occurs only along two coordinates instead of allwhere the difference in Fermi energies between the two states
is parameterized using δ = (EFf − EFi )/EFf . For instance,
Ni = Nf (1 − δ)3 , etc. Figures 7 and 8 show the normalized
rate of Raman scattering between imbalanced Fermi clouds.
At finite temperature, figure 7 shows SIP versus δ for κ =0.5, demonstrating that an imbalance enhances suppression.
Increasing the cloud imbalance enhances blocking because
the range of initial states is increasingly restricted to lower
energies. At low temperature, the effect can be dramatic,
allowing for complete blocking at δ  3/4.
The zero-temperature limit of Raman scattering is16
SIP (κ(α), δ) = (κ(α) + b − 1) 1 −(b + B)χ (−B/b)5π16(1 + b − κ(α))χ (κ(α) + B)/b6 ,
(24)
+
5π
6

Nearly exclusive decay to nf could be achieved by exciting along a weaktransition, such as a narrow line, or a pair of magnetic substates with a weak
Clebsch–Gordan coefficient.

6

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 015301

B Shuve and J H Thywissen

scattering can be achieved with Raman scattering between
internal states for large population imbalances and and EF >4ER . Our calculations should aid experimental efforts toobserve this fundamental quantum optical effect.

six coordinates7 . Both schemes are successful at enhancing
the expected signal in realistic experimental scenarios: atκ = 0.25 and T = 0.1TF , we find MLS = 0.40, MFB = 0.32
for σ/RF = 0.50, and MIP = 0.07 for δ = 0.90. The dramatic
suppression that seemed like a distant experimental prospect insection 3.1 (since it required a thermalized excited-state atom)
is feasible using a Raman light scattering scheme.

Acknowledgments
We thank T Busch for additional notes regarding [4]. We also
thank B DeMarco, D F V James, C Salomon and J I Cirac forstimulating discussion, and the Toronto ultra-cold atoms groupfor discussion and proof reading. This work was supported by
NSERC and the Canadian Institute for Advanced Research.
JHT thanks the MIT-Harvard Center for Ultracold Atoms fortheir hospitality during the preparation of this manuscript.

5. Experimental realization and conclusion
To realize a Fermi energy that is 1/κ 2 times the recoil energy,
the mean trap strength must be
h̄ω =

ER
.
κ 2 (6N )1/3

(25)

For 104 40 K atoms, ω = 2π × 860 Hz at EF = 4ER . For
106 6 Li atoms, ω = 2π × 1620 Hz at EF = 4ER .
In order to avoid multiple scattering, the atomic sample
must have a low optical density. Consider a cloud of N
fermions at T = 0 in a cylindrically symmetric trap, whose
eccentricity is  ≡ ωz /ω⊥ . A Thomas–Fermi profile has aresonant optical density bounded by
ODmax = max{ 1/3 ,  −2/3 }

3
4

EF
ER

2

ER
.
h̄ω̄

Appendix: angle-averaged results for zerotemperature
Substitution of (15) into (13) yields an analytic expression forzero-temperature angle-averaged scattering in the LS case:⎧√1⎪
1 − κ 2 − 1)
3 {512(⎪2520πκ
⎪2⎪⎪+
315κ(16κ+ 3) sin−1 (κ)
⎪
√
⎪
⎪
2
2
4
⎪
⎨ + κ [2κ 1 − κ 2 (232κ2− 1036κ
MLS (κ) =(A.1)√ + 2157)⎪
2 − 3456]}
⎪
1
−
κ
for
κ

1
+
2767
⎪
⎪
⎪
⎪
⎪
1 − [256(27κ 2 + 4)⎪⎪⎩
for κ > 1.
− 945π κ]/(5040π κ 3 )This quantity is plotted as a dashed line in figure 4.
A similar expression can also be found for the FB case in
the tightly focused limit:
√
⎧ 1
2
2
⎪
⎪ 210πκ 3 {κ [2 1 − κ √(131⎪
2 2⎪
263 1 − κ 2⎪⎨ − 32κ )κ + √2D− 336] + 64( 1 − κ 2 − 1)
(κ) =
MLS
⎪
3
−1
⎪
for κ  1
⎪ + 105(4κ + κ) sin (κ)}⎪⎪⎩
21κ(32κ−5π)+128
1−
for κ > 1.
420πκ 3(A.2)
The difference between (A.2) and (A.1) is plotted as a dashedline in the inset of figure 6.

(26)

Combining with (25) and the constraint that OD < 1 requiresthat N be on the order of unity, which is clearly incompatible
with a Pauli blocking experiment. Quantitative predictions for
spontaneous emission and for light scattering at resonance will
require treatment of multiple scattering.
Scattering with off-resonant light can avoid thiscomplication. Detuning approximately N 1/6 line widths awayfrom resonance can achieve OD < 1. Our focus on net
scattering rates rather than on line shape is partially motivatedby this limitation.
One could use a variety of signatures to search forblocking effects in the lab. Because fewer than one photon
per atom can be scattered while remaining in the perturbative
limit, angle-resolved experiments that measure only a fractionof the total scattered light may be difficult. However, a direct
measure of the integrated rate M, rather than S(α), is possible
through absorption imaging. Another measure of M would
be an optical pumping experiment, in which the efficiencyof pumping into an occupied Fermi sea is reduced due toblocking effects. In this case the blocking effect would berecorded in atomic populations, circumventing lensing effectsof the detuned absorption beam by the degenerate cloud.
In summary, we have presented two light scattering
scenarios in which Pauli blocking is strengthened. Including
the effects of inhomogeneous trapping, finite temperature andwithout assuming rotational symmetry, we make quantitative
predictions for both angle-resolved and angle-averaged
signatures. Pauli blocking effects can be enhanced by focusing
the excitation beam. Dramatic suppression of incoherent

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