Exploring transport in a Tonks
Girardeau gas
Carlo Sias
www.quantumoptics.eu
£££: EPSRC, University of Cambridge, Herchel-Smith Fund, ERC
Transport – a very general subject!
I
U
potential difference U
continuously accelerates
electrons
Open system:
energy is continuously transferred
into the system
Transport experiments in cold atoms
Ketterle group 2000
A simple transport experiment
k
Three
dimensions
-k
One
dimension
-2ħk
0
+2ħk
Scattered atoms with
spherical (s-wave) symmetry
Weiss group 2006
Other transport experiments: ETH, LENS, NIST, Orsay, Pisa, Stanford,, ...
Transport in solid state: 1-dimension
Transistors for transparent
and flexible electronics
Solar cell design
Spin-charge separation
What is a one-dimensional gas?
What is a one-dimensional gas?
• transverse degrees of freedom are frozen out
Conditions for 1D
w
kBT < hw┴
Bosons: m < hw┴
Fermions: EF =N hwz < hw┴
1-dimensional Bose gas
• Exactly solvable many-body problem! (Lieb-Liniger)
• The many-body quantum state is characterized by a
single parameter
mg1D
 2 n1D


 
2
 
(

g

(
x

x
)
(
r
)

E

(
r
)

r
)


(
r
)
;
mc
m

1
D
i
j

Bose
i
2


 i 2m xi i , j 
 i 1... N
2
2

weakly interacting Bose gas
crossover
mg
  2 1D
 Tonks-Girardeau
n1D
gas
 Bose (r )   Fermi (r )
(“Fermionized Bosons”) k   n  mc / 
F
Excitations in a weakly interacting Bose gas
 «1
Bogoliubov spectrum
Excitations in a Tonks gas
→∞
Two branches of excitations: “particle” excitations and “hole” excitations
q
q
-kF
kF
k
-kF
kF
k
Generating tight confining potentials
Induced electric dipole potential:
V
ac polarizability of the atom
wL  w A
Two options:
„red detuned“
wL
1
 
2
E
2
electric field of the laser
wL  w A
„blue detuned“
wA
Optical lattice
Energy scale:
 2k 2
Erec 
2m
l/2
Hybrid optical/magnetic trap
Vertical confinement:
purely magnetic
Other experiments in 1D: ENS, ETH, LENS, Mainz, MIT, NIST, Orsay, Penn State, Rice, Vienna, Innsbruck …
Hybrid optical/magnetic trap
Experimental parameters:
Atoms: 87Rb (bosons)
Wavelength of lattice: 764 nm
wx= wy ≤ 2 65 kHz (optical lattice)
wz= 2 39 Hz (magnetic trap)
N<50 per tube
0.5 <  < 10
Vertical confinement:
purely magnetic
Other experiments in 1D: ENS, ETH, LENS, Mainz, MIT, NIST, Orsay, Penn State, Rice, Vienna, Innsbruck …
Generation of spin impurities
z
F=2
ΔE(z)
F=1
mF:
-2
-1
0
1
2
Generation of spin impurities
z
wRF  mB(z)
F=2
quick transfer
(RF /2-pulse, 200 ms)
F=1
mF:
-2
-1
0
1
2
Generation of spin impurities
z
wRF  mB(z)
F=2
F=1
mF:
-2
-1
0
1
2
Generation of spin impurities
• width of impurity wavez packet:
2.5 mm (≈ 3 atoms)
• same transverse confinement:
propagation of impurities is
purely one-dimensional
• same scattering lengths:
a-1,-1 ≈ a-1,0 ≈ a0,0
• accelerated impurity breaks
integrability of the 1D Bose
gas → interesting dynamics
In situ detection of the spin wave packet
F=2 imaging only
F=2
F=1
In situ detection of the spin wave packet
F=2 imaging only
F=2
F=1
Distance z [mm]
Time evolution
g
• strong interaction-induced dynamics
• significant back action of the impurity onto the majority component
• open quantum system: impurity atoms can transfer continuously
energy into trapped component by collisions
S. Palzer, C. Zipkes, C. S., Michael Köhl, PRL 103, 150601 (2009)
Center-of-mass motion of the impurities
Distance z [mm]
S. Palzer, C. Zipkes, C. S., Michael Köhl, PRL 103, 150601 (2009)
Center-of-mass motion of the impurities
Tonks gas
Distance z [mm]
weakly interacting Bose gas
ballistic
S. Palzer, C. Zipkes, C. S., Michael Köhl, PRL 103, 150601 (2009)
Center-of-mass motion of the impurities
Tonks gas
weakly interacting Bose gas
ballistic
part of the
impurities
have already
left the gas
S. Palzer, C. Zipkes, C. S., Michael Köhl, PRL 103, 150601 (2009)
Release measurement
tdelay
S. Palzer, C. Zipkes, C. S., Michael Köhl, PRL 103, 150601 (2009)
Release measurement
tdelay
S. Palzer, C. Zipkes, C. S., Michael Köhl, PRL 103, 150601 (2009)
Release measurement
tdelay
S. Palzer, C. Zipkes, C. S., Michael Köhl, PRL 103, 150601 (2009)
Release measurement
tdelay
Simple model
Every collision resets the impurity’s velocity
to 0, then gravity accelerates again
Mean time between collision events:
tcoll  2 /( n1D ( z) g )
Time delay accumulated: total number of collisions x time delay per collision
z0
2
t   dz n1D ( z ) ( z ) tcoll
R
S. Palzer, C. Zipkes, C. S., Michael Köhl, PRL 103, 150601 (2009)
“Far field” distribution of the impurities
“Far field” distribution of the impurities
“Far field” distribution of the impurities
“Far field” distribution of the impurities
scattered
 =3
 =5
unscattered
 =7
c)
b)
a)
S. Palzer, C. Zipkes, C. S., Michael Köhl, PRL 103, 150601 (2009)
“Far field” distribution of the impurities
scattered
 =3
 =5
unscattered
 =7
c)
b)
a)
• Fermionization?
• Enhancement of multiple collision events due to strong interactions?
•?
S. Palzer, C. Zipkes, C. S., Michael Köhl, PRL 103, 150601 (2009)
How to control the impurities in the gas?
• Problem: once created, the impurity dynamics cannot be
controlled
How to control the impurities in the gas?
• Problem: once created, the impurity dynamics cannot be
controlled
FORGET ABOUT SPIN IMPURITIES,
WE NEED A NEW STRATEGY!
Impurities in a Bose gas: a new approach
Quantum degenerate atoms
Ultracold trapped ions
• Macroscopic quantum system
at nano Kelvin temperature
• Collective quantum states of 106 particles
• Very long coherence times
• Single particle detection and
manipulation
• Pristine source for quantum information
processing & precision spectroscopy
Hybrid quantum system
A new approach to
• Ultracold collisions & quantum chemistry
• Quantum information processing and decoherence
• Quantum many-body physics with impurities
related work with trapped ions @ MIT, Innsbruck/Ulm, Uconn, Weizmann,UCLA,..[free ions: Rice, Michigan, Durham, Maryland, ... ]
Atom-ion hybrid trap
Ultracold atom-ion collisions
Ion trap
(level spacing  5 mK)
Bose-Einstein condensate
T  100 nK, trap depth  3 mK
Ultracold atom-ion collisions
Ion trap
(level spacing  5 mK)
Bose-Einstein condensate
T  100 nK, trap depth  3 mK
Measured atom loss cross section:
sloss = (2.2±0.2)·10-13 m2
C. Zipkes, S. Palzer, CS, Michael Köhl, Nature, 464 388(2010)
Determine ion temperature from sloss  sel(kBT) +  (2 kBT/mw2)
T  2 mK
(almost Doppler temperature)
Sympathetic cooling of an ion in a BEC
• Rapid cooling
• First demonstration of
cooling an object
different from other
neutral species using
ultracold atoms
C. Zipkes, S. Palzer, CS, Michael Köhl, Nature, 464 388(2010)
• Ultimate cooling limits
need to be explored
with a different
technique (eg. Raman
spectroscopy)
Atom-ion reaction processes
• Elastic collisions
ion changes vibrational quantum state, atom heats up
• Charge-exchange collisions
Yb+ + Rb → Yb + Rb+
Yb+ + Rb → (YbRb)+
C. Zipkes, S. Palzer, L. Ratschbacher, CS, Michael Köhl, ArXiv:1005.3846
A single impurity as a local probe
C. Zipkes, S. Palzer, L. Ratschbacher, CS, Michael Köhl, ArXiv:1005.3846
Summary
• Quantum transport in a strongly interacting Bose gas
with an accelerated impurity
• High resolution tomography
• Strong dynamics and back action of the impurity
• Far field distribution: effects of fermionization?
• New approach to study impurities: single trapped ion in a
BEC
Thanks!
PI
M. Köhl
Ion & BEC:
S. Palzer, C. Zipkes, L. Ratschbacher, H. Meyer, M. Steiner. C.S.
Fermi gas:
M. Feld, B. Fröhlich, E. Vogt, K. Beck
(first Fermi gas in the UK since Nov 2009)
Funding: EPSRC, ERC, Univ. of Cambridge, Herchel-Smith Fund
Dynamic structure factor
Scattering rate of an impurity (Fermi’s golden rule):
   dq dw S (w , q)  ( (ki )   (k f )  w (q))
S(q,w): dynamic structure factor
ki , kf : initial and final momentum of the impurity
w(q): excitation spectrum of the gas
E   dq dww S (w , q)  ( (ki )   (k f )  w (q ))
Collision rate and energy dissipation
For equal masses:
impurities move collisionless through a superfluid and a Tonks-Girardeau
gases for v<c.
Collision rate (v>c)
  2 n1D
weakly interactin g Bose gas
 2 2
m
a
v

1D

 n1D ln  ki / k F  1  Tonks  Girardeau gas
 2ma12D k F  ki / k F  1 
Energy dissipation (v>c)
4
 2 2 n


c


1D 

v
1




2

 ma1D   v  

E
2
 2 n1D v
 ma 2
1D

weakly interactin g Bose gas
Tonks  Girardeau gas
for heavy impurities: Astrakharchik & Pitaevskii, Davis et al., ...
constant force
 50% of gravity