Download Three-body Interactions in Cold Polar Molecules

Three-body Interactions in Cold Polar
Molecules
H.P. Büchler
Theoretische Physik, Universität Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation der Österreichischen
Akademie der Wissenschaften, Innsbruck, Austria
In collaboration with:
A. Micheli, G. Pupillo, P. Zoller
Three-body interactions
Many-body interaction potential
- Hamiltonians in condensed matter are effective Hamiltonains
after integrating out high energy excitations
1 !
1!
V (ri − rj ) +
W (ri , rj , rk ) + . . .
Veff ({ri }) =
2
6
i!=j
i!=j!=k
two-particle
interaction
three-body
interaction
Application
- Pfaffian wave function of fractional
quantum Hall state (More and Read, ‘91)
- exchange interactions in spin systems:
microscopic models exotic phases
(Moessner and Sondi, ’01, Balents et al., ’02,
Moutrich and Senthil ‘02)
- string nets: degenerate Hilbertspace for
loop gases (Fidowski, et al, ‘06)
Route towards
exotic and topological
phases?
Three-body interactions
Extended Bose-Hubbard models
- hardcore bosons
H = −J
!
b†i bj
!ij"
1 !
1!
Uij ni nj +
Wijk ni nj nk .
+
2
6
hopping energy
i#=j
i#=j#=k
two-body interaction
Goal
- large interaction strengths
- independent control of two- and
three-body interaction
three-body interaction
1D phase diagram
µ/W
n = 2/3
n = 1/2
Realizable with
polar molecules
n = 1/3
J/W
Polar molecules
Hetronuclear Molecules
+
- electronic excitations
∼ 10 Hz
15
- vibrational excitations
∼ 1013 Hz
- rotational excitations
∼ 10 Hz
10
- electron spin
- nuclear spin
Polar molecules in the
electronic, vibrational, and
rotational ground state
- permanent dipole
moment: d ∼ 1−9 Debye
- polarizable with static electric
field, and microwave fields
−
dipole
moment
ea0 ≈ 2.5Debye
Strong dipole-dipole
interactions tunable with
external fields
d1 d2
(d1 r) (d2 r)
V (r) = 3 − 3
r
r5
Interaction energies
Particles in an optical lattice
- lattice spacing
a = λ/2 ∼ 500nm
- size of Wannier function
ah.o ∼ 0.2a
2!2 π 2
- recoil energy Er =
mλ2
Pseudo-potential
- dominant interaction in
atomic gases
on-site
interaction
nearest-neighbor
interaction
U ∼ 0.5Er
present but
small
U ∼ 0.5Er
U1 ∼ 10−3 Er
Magnetic dipole moment
- Chromium atoms with
m ∼ 6µB
Electric dipole moment
- LiCs hetronuclear molecule
d ∼ 6.5Debye
2
2
- increased by factor 1/α ∼ (137)
U ∼???
(a/ah.o )3 U1
U1 ∼ 30Er
Polar molecules
molecular ensembles
(quantum memory)
AMO- solid state interface
- solid state quantum processor
- molecular quantum memory
(P. Rabl, D. DeMille, J. Doyle, M. Lukin,
R. Schoelkopf and P. Zoller, PRL 2006)
Cooper Pair Box
(superconducting qubit)
ZZ
XX YY
Spin toolbox
- polar molecules with spin
- realization of Kitaev model
(A. Micheli, G. Brennen, P. Zoller,
Nature Physics 2006)
Polar molecules
Experimental status
Raman laser /
spontaneous emission
- Polar molecules in the rotational and
vibrational ground state
- cooling and trapping techniques
beeing developement:
- cooling of polar molecules:
e.g. stark decelerator
D. DeMille, Yale
J. Doyle, Harvard
G. Rempe, Munich
G. Meijer, Berlin
J. Ye, JILA
- photo association
see J. Ye’s talk
(all cold atom labs)
- bosonic molecules with closed
electronic shell, e.g., SrO, RbCs, LiCs
rotational and vibrational
ground state
Polar molecules
Crystalline phases
- long range dipole-dipole
interaction
- interaction energy exceeds
kinetic energy
Three-body interaction
- extended Hubbard models
- tunable three-body interaction
Polar molecule
rotation of
the molecule
Low energy description
- rigid rotor in an electric field
: angular momentum
dipole
moment
: dipole operator
N =2
Accessible via microwave
- anharmonic spectrum
- electric dipole transition
N =1
N =0
}
- microwave transition frequencies
- no spontaneous emission
Interaction between polar molecules
Hamiltonian
kinetic
energy
Without external drive
- van der Waals
attraction
Static electric field
- internal Hamilton
- finite averaged dipole moment
trapping
potential
rigid
rotor
electric
field
interaction
potential
Dipole-dipole interaction
Dipole-dipole interaction
- anisotropic interaction
- long-range
repulsion
attraction
- Born-Oppenheimer
valid for:
attraction
Instability in the
many-body system
- collaps of the system for
increasing dipole interaction
- roton softening
- supersolids?
(Goral et. al. ‘02, L. Santos et al. ‘03, Shlyapnikov ‘06)
Stability:
- strong interactions
- confining into 2D
by an optical lattice
z
a⊥
{
confining
potential
oscillator
wavefunction
Stability via transverse confining
Effective interaction
+1.5
V(R,z)l!3/D
- interaction potential with
transverse trapping potential
2
!
"
1
z
V (r) = D 3 − 3 5 +
r
r
mωz2 2
2
z/l!
z
1
0
- characteristic
length scale
0
- potential barrier:
larger than kinetic energy
{1.5
0.5
1
0
1
attempt frequency
Tunneling rate:
- semi-classical rate
(instanton techniques)
- Euclidean action of the
SE = h̄
instanton trajectory
numerical
factor:
Dm
h̄2 a⊥
"2/5
R/l!
3
kinetic
energies
Γ = A exp (−SE /h̄)
!
2
C
bound
states
4
Crystalline phase
Hamiltonian
interaction
strength:
- polar molecules confined into a
two-dimensional plane
Dm
Eint
= 2
rd =
Ekin
! a
rd = 1 − 300
Kosterlitz-Thouless
transition
First order melting
(Kalida ‘81)
Quantum melting
(H.P. Büchler, E. Demler, M. Lukin, A. Micheli,
G. Pupillo, P. Zoller, PRL 2007)
- indication of a first order transition
- critical interaction strength
rd ≈ 20
Three-body interactions
1 !
1!
V (ri − rj ) +
W (ri , rj , rk ) + . . .
Veff ({ri }) =
2
6
i!=j
i!=j!=k
H.P. Büchler, A. Micheli, and P. Zoller
cond-mat/0703688 (2007).
Single polar molecule
shifted away by
external DC/AC
fields
Static electric field
- along the z-axes
- splitting the degeneracy of the first
excited states degeneracy
- induces finite dipole moments
|e, 0!
|e, 1!
|e, −1"
∆
Ω
dg = !g|dz |g"
de = !e, 1|dz |e, 1"
|g!
Mircowave field
- coupling the state |g! and
∆
Ω
|e, 1!
: detuning
: rabi frequency
- restrict to two states
- ignore influence of |e, −1"
- rotating wave approximation
- anharmonic spectrum
- electric dipole transition
- microwave transition frequencies
- no spontaneous emission
Many-body Hamiltonian
Many-body Hamiltonian
{
{
! p2
!
! (i)
i
stat
ex
H=
+
Vtrap (ri ) +
H0 + Hint
+ Hint
2m
i
i
i
- external potentials:
- dipole-Dipole interaction
- restriction to the two
internal states:
- trapping potential
- optical lattices
|g!i |e, 1!i
Two-level System
- rotating wave approximation
(i)
H0 =
1
2
!
∆
Ω
Ω
−∆
- two eigenstates
"
- two-level system in an effective
magnetic field
= hSi
|+!i = α|g!i + β|e, 1!i
|−"i = −β|g"i + α|e, 1"i
and energies
!
E± = ± Ω2 + ∆2 /2
Dipole-dipole interaction
Microwave photon exchange
D = |!e, 1|d|g"|2 ≈ d2 /3
-
H
ex
int
" + −
#
1!D
+ −
=−
ν(ri − rj ) Si Sj + Sj Si
2
2
i!=j
dipole-dipole
interaction
Induced dipole moments
√
- ηd,g = de,g / D
H
stat
int
1 − cos θ
ν(r) =
r3
Pi = |g!"g|i
1!
Dν (ri − rj ) [ηg Pi + ηe Qi ] [ηg Pj + ηe Qj ]
=
2
i!=j
Qi = |e, 1!"e, 1|i
Born-Oppenheimer potentials
Effective interaction
(i) diagonalizing the internal Hamiltonian
for fixed interparticle distance {ri } .
!
(i)
stat
ex
H0 + Hint
+ Hint
i
(ii) The eigenenergies E({ri })
describe the Born-Oppenheimer
potential a given state manifold.
R0
(iii) Adiabatically connected to the
groundstate
|G! = Πi |+!i
“weak” dipole interaction
√
D
= R03 " a3
∆ 2 + Ω2
interparticle
distance
ri − rj
Born-Oppenheimer potential
First order perturbation
ex
stat
E (1) ({ri }) = !G|Hint
+ Hint
|G"
!
- |G! =
(α|g!i + β|e, 1!1 )
-
i
E
(1)
1 !
Dν(ri − rj )
({ri }) = λ1
2
i!=j
dipole-dipole
interaction:
1 − 3 cos θ
Veff (r) = λ1
r3
Dimensionless coupling parameter
-
!
λ1 = α ηg + β ηe
2
2
"2
- tunable by the external electric field
and the ratio Ω/∆ .
− α2 β 2
dE/B
- for a magic rabi frequency the
dipole-dipole interaction vanishes
λ1 = 0
Born-Oppenheimer potential
Second order perturbation
E (2) ({ri })
!
=
k!=i!=j
+
!
i!=j
2
|M |
√
D2 ν (ri − rk ) ν (rj − rk )
∆ 2 + Ω2
2
|N |
2
√
[Dν (ri − rj )]
∆ 2 + Ω2
: repulsive two-body
interaction
Matrix elements
-
M = αβ
!"
#
α ηg + β ηe (ηe − ηg ) + (β − α )/2
!
"
2
2 2
N = α β (ηe − ηg ) + 1
2
: three-body
interaction
2
2
2
$
Effective Hamiltonian
Effective interaction
1 !
1!
V (ri − rj ) +
W (ri , rj , rk )
Veff ({ri }) =
2
6
i!=j
i!=j!=k
- two-body interaction
2
V (r) = λ1 D ν (r) + λ2 DR03 [ν (r)]
- three-body interaction
W (r1 , r2 , r3 ) = γ2 R03 D [ν(r12 )ν(r13 ) + ν(r12 )ν(r23 ) + ν(r13 )ν(r23 )]
- validity is restricted to
D
√
= R03 " a3
∆ 2 + Ω2
interparticle
distance
(i) transverse confining
into 2D
(ii) vanishing dipole-dipole
interaction
+1.5
V(R,z)l!3/D
z/l!
1
0
0
{1.5
0.5
1
0
1
2
R/l!
3
4
Bose-Hubbard
model
Hubbard model
Extended Bose-Hubbard models
- hardcore bosons
H = −J
!
b†i bj
!ij"
hopping energy
- interaction parameters
for strong optical lattices
1 !
1!
Uij ni nj +
Wijk ni nj nk .
+
2
6
i#=j
two-body interaction
Uij = V (Ri − Rj )
i#=j#=k
three-body interaction
Wijk = W (Ri , Rj , Rk )
Polar molecule: LiCs:
- dipole moment
d ≈ 6Debye
- hopping energy
J/Er ∼ 0 − 0.5
- lattice spacing:
- nearest neighbor
interaction:
λ ≈ 1000 nm
Er ≈ 1.4 kHz
U/Er ∼ 30
3
W/Er ∼ 30 (R0 /aL )
Hubbard model
Three-body interaction
- next-nearest neighbor terms
Wijk = W0
!
6
a
|Ri − Rj |3 |Ri − Rk |3
"
+ perm
Supersolids on a triangular lattice
H=
(1)
!
!
1
†
Uijonnithe
njtriangular lattice
−J
b bj + bosons
Supersolidi hardcore
2
!ij"
i#=j"
Stefan Wessel(1) and Matthias Troyer(2)
Institut für Theoretische Physik III, Universität Stuttgart, 70550 Stuttgart, Germany and
(2)
Theoretische Physik, ETH Zürich, CH-8093 Zürich, Switzerland
(Dated: May 18, 2006)
1
Uij ∼ the phase diagram
: static
electric
We determine
of hardcore
bosonsfield
on a triangular lattice with nearest neighbor
3
− j|
repulsion, paying |i
special
attention to the stability of the supersolid phase. Similar to the same model
PACS numbers:
Quantum Monte Carlo simulations
1
Next and
to the
widely
Wessel
Troyer,
PRLobserved
(2005) superfluid and Bosecondensed
broken U (1) symmetry and “crysMelko
et al.,phases
PRB,with
(2006)
talline” density wave ordered phases with broken transsymmetry, the supersolid phase, breaking both
- lational
supersolid
close to half filling
the U (1) symmetry and translational symmetry has been
strong
nearest
neighbor
aand
widely
discussed
phase
that is hard to find both in
interactions
experiments
and in theoretical models. Experimentally,
evidence
for 1/2
a possible supersolid phase in bulk 4 He has
n=
recently been presented [1], but the question of whether
! 10
a trueU/J
supersolid
has been observed is far from being set[2, 3],
leavingnext-nearest
the old question of supersolid behavior
- tled
stable
under
in translation invariant systems [4, 5] unsettled for now.
neighbor
interactions
More precise statements for a supersolid phase can be
made for bosons on regular lattices. It has been proposed that such bosonic lattice models can be realized
0.8
PS
solid !=2/3
PS
0.6
supersolid
!
505298 v1 11 May 2005
on a square lattice we find that for densities ρ < 1/3 or ρ > 2/3 a supersolid phase is unstable and
the transition between a commensurate solid and the superfluid is of first order. At intermediate
fillings 1/3 < ρ < 2/3 we find an extended supersolid phase even at half filling ρ = 1/2.
0.4
superfluid
PS
solid !=1/3
PS
0.2
0
0
0.1
0.2
t/V
0.3
0.4
0.5
FIG. 1: Zero-temperature phase diagram of hardcore bosons
One-dimensional model
H = −J
!
b†i bi+1
+W
i
!
i
ni−1 ni ni+1 − µ
!
ni
next-nearest
neighbor interactions
+ Hn.n.n.
i
Bosonization
Critical phase
- hard-core bosons
- instabilities for densities:
n = 2/3 n = 1/2 n = 1/3
- quantum Monte Carlo simulations
(in progress)
- algebraic correlations
- compressible
- repulsive fermions
µ/W
Solid phases
n = 2/3
- excitation gap
- incompressible
- density-density correlations
n = 1/2
!∆ni ∆nj "
- hopping correlations (1D VBS)
n = 1/3
J/W
!b†i bi+1 b†j bj+1 "
Conclusion and Outlook
Polar molecular crystal
- reduced three-body collisions
- strong coupling to cavity QED
- ideal quantum storage devices
Lattice structure
- alternative to optical lattices
- tunable lattice parameters
- strong phonon coupling: polarons
Extended Hubbard models
- strong nearest neighbor interaction
- three-body interaction
Novel quantum matter
- supersolid phases
- string nets?