73
Collective excitations of ultracold molecules on an optical lattice Roman Krems University of British Columbia
Collective excitations of ultracold molecules on an optical lattice
Roman Krems University of British Columbia
Collective excitations of ultracold molecules�trapped on an optical lattice�
Sergey Alyabyshev Chris Hemming Felipe Herrera Jie Cui Marina Li9nskaya Jesus Perez Rios Ping Xiang Alisdair Wallis
Funding:
Peter Wall Ins9tute for Advanced Studies
Roman Krems�
Zhiying Li, now at Harvard University Timur Tscherbul, now at Harvard University
This talk
1. Ultracold chemistry – a new exciting research field
2. New physics with ultracold molecules in an optical lattice
12 Nobel prizes to 27 scientists for research of low temperature phenomena since 1913
1997 - for the development of methods to cool and trap atoms with laser light
2001 - for the achievement of Bose-Einstein condensation in dilute gases of alkali metal atoms, and for early
fundamental studies of the properties of the condensates
12 Nobel prizes to 27 scientists for research of low temperature phenomena since 1913
1997 - for the development of methods to cool and trap atoms with laser light
2001 - for the achievement of Bose-Einstein condensation in dilute gases of alkali metal atoms, and for early
fundamental studies of the properties of the condensates
20XX - for Ultracold Chemistry
10-8
10-7
10-6
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10-4
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100
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Temperature scale (Kelvin)
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Temperature scale (Kelvin)
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Temperature scale (Kelvin)
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Temperature scale (Kelvin)
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Coldest T in
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Magnetic trap
middle of the trap
Mag
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ield
More delicate methods: evaporative coolingEvaporative cooling
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Ultracold chemistry – new regime of chemistry
• controlled chemical reactions
• quantum effects in chemistry
• detailed mechanisms of chemical reactions
• role of individual ro-vibrational energy levels in determining chemical reactivity
Possibility to study
See “Cold Controlled Chemistry”: R. V. Krems, PCCP 10, 479 (2009)
Ultracold chemistry – new regime of chemistry
• effects of quantum statistics and many-body physics on chemical reactions
• effects of tunable fine and hyperfine interactions on chemical reactions
• effects of external space symmetry on chemical reactions
Possibility to study
See “Cold Controlled Chemistry”: R. V. Krems, PCCP 10, 479 (2009)
Collective excitations of molecules in an optical lattice
It has now become possible to create dense ensembles of diatomic molecules, both polar and non-polar, at nanoKelvin temperatures
Ultracold molecules on optical lattices = molecular crystals with unusual properties:
Intermolecular interactions are very weak, much weaker than the energy of rotational
splitting in molecules
Molecules are held in the crystal by optical field forces, not intermolecular interactions
Is rotational excitation of a molecule a single-particle or collective excitation?
What can we do with molecules on a lattice that cannot be done with conventional crystals?
Frenkel exciton
φn = |00〉1|00〉2...|10〉n|00〉n+1...|00〉N
ψ =∑n
Cnφn
Frenkel exciton
φn = |00〉1|00〉2...|10〉n|00〉n+1...|00〉N
ψk =∑n
eik·rn√Nφn
Dispersion Curves!
-20
-10
0
10
20
E(k
) (k
Hz)
0 1 2 3 k a
-100
-75
-50
-25
0
25
50
E(k
) (k
Hz)
0 1 2 3 k a
-25
-20
-15
-10
-5
0
5
10
15
20
E(k
) (
in u
nit
s of
10
-6B
)
!
" , #
!
"
#
Negative effective mass => ! negative refraction of EM field!
Rotational excitons are controllable…
-40
-20
0
20
40
E(k
) (k
Hz)
0 1 2 3k a
-120
-80
-40
0
40
80
E(k
) (k
Hz)
-90
-60
-30
0
30
60
E(k
) (
kHz)
0 1 2 3k a
-40
-20
0
20
40
60
E(k
) (
kHz)
E E
α, β
γ
α
β
γ
xx
Excitons are sensitive to impurities …
Exciton – impurity interactions can be controlled!
Impurities!
Pure Exciton Hamiltonian:
Impurities!
Scatterer with the strength = difference in transition energies:
Breaks translational symmetry Mixes states with different k
One impurity:
-60
-30
0
30
60
∆Eeg
(kH
z)
0 1 2 3 4ε (kV/cm)
1.0
1.2
1.4
1.6
1.8
2.0∆E
eg (
×10
4 MH
z)
-30 -15 0 15 30ε − ε 0 (mV/cm)
100
102
104
106
108
σ 2D
(Å)
LiCs
LiRb
cross sections for |k|! !/a, in 3D and 2D, as
"3D(k, V0) =2!!2/|m!|Tk + E(3D)
b
, (5a)
"2D(k, V0) =4!2/k
!2 + ln2
!E
(2D)b (!"Tk)
Tk(!"E(2D)b )
" , (5b)
where E(3D)b = (2/!"V0/!)2! and E(2D)
b = !/ [exp(4!/!V0)" 1]. In 3D, the bound state
exists only if V0 > 2!/!. Resonant enhancement of the scattering cross section occurs for
values of V0 that support a shallow bound state (Eb # T $ 0). As a result of the negative
e"ective mass of the exciton, the bound state that leads to resonance is produced by a
repulsive potential. Equations (??) are derived in the approximation !Jn,0 = 0. Including
!Jn,0 in the calculation leads to a shift of the positions of the resonance and the resonant
enhancement of the scattering cross section at a slightly di"erent value of V0 [? ].
FIG. 2: (color online) (a) Excitation energies !Eeg for transitions |0, 0% $ |1,M% with M = 0
(upper curve) and M = ±1 (lower curve) vs electric field for three polar molecules. (b) Expanded
view of the encircled area in panel (a). (c) Exciton-impurity 2D scattering cross sections for
|k|a = 4 & 10"5 (solid line), |k|a = 4 & 10"3 (dashed line), and |k|a = 4 & 10"2 (dotted line).
a = 400 nm and E0 = 3228.663 V/cm
Experiments with optical lattices allow for controlled deposition of ultracold particles at
di"erent lattice sites, which can be used to generate exciton-impurity systems with con-
trolled spatial distributions of impurities. Another advantage of optical lattices for studying
exciton dynamics is the absence of phonons [? ]. Therefore, for low exciton density, one
can use scattering by impurities to fully control the dynamics of exciton wavepackets. A
generalization of Hamiltonian (??) describes the dynamics of an exciton in the presence of
Ni substitutional impurities at positions in in the lattice. This Hamiltonian can be written
as H = H0 + W + V , with the corresponding matrix elements in the basis of free-exciton
states |#k% given by
'H0%q,k = E(k)#k,q, (6a)
'W %q,k =2!J(a)
Nmol(cosq · a + cosk · a)
Ni#
in=1
ei(q"k)·in , (6b)
5
Exciton – impurity Hamiltonian matrix!
!V "q,k =V0
Nmol
Ni!
in=1
ei(q!k)·in , (6c)
where V = V0, and !J(a) = !Jn,n!1. Here, we neglect the fast-decaying terms !Jn,m
with |n # m| > 1. The terms V and W correspond to diagonal and o"-diagonal disorder,
respectively, in the site representation.
FIG. 3: (color online) Probability density |!(x)|2 (solid line) of a lattice eigenstate near the top of
the energy band, for a 1D array of 1000 LiCs molecules, with 1% of LiRb impurities (dots). Panels
correspond to di"erent values of V0: (a) V0 = 0, (b) V0/h = 21 kHz, and (c) V0/h = 29 kHz.
When an external electric field is such that V = 0, exciton-impurity scattering occurs
only due to the di"erence in dipole moments between host and impurity molecules. The
eigenstates of the corresponding Hamiltonian are localized wavepackets in real space. Figure
3 shows the probability density of a particular eigenstate of Hamiltonian (6) near the top
of the energy band, for di"erent values of V0. We consider a 1D array of LiCs separated by
400 nm, with a random distribution of LiRb impurities (d0 = 4.165 Debye, 2Be = 13.2268
GHz [? ]), which gives !J(a)/h = #6.89 kHz. Due to the negative e"ective mass of free-
exciton states, high energy eigenstates are dominated by free-exciton states with k $ 0.
These eigenstates are localized (Fig. 3a). Delocalization of these states can be achieved by
applying an electric field so that V0 $ #4!J(a) (see Fig. 3b). In this case, for a given
k, the matrix elements !V "q,k and !W "q,k cancel for q $ k, which suppresses the coupling
between the corresponding free-exciton states. Localized eigenstates with di"erent energies
become delocalized at di"erent values of V0 because !W "q,k is wavevector dependent. The
wavepackets are localized for values of V0 that do not balance the e"ect of !J(a) (Fig. 3c).
Microwave photons with linear polarization can be used to generate a rotational excita-
tion. We consider an excitation generated in a lattice with impurities in the presence of
an electric field that corresponds to V0 = 0. The electric field can then be tuned to intro-
duce a repulsive or attractive potential V0. This can be used to modify the dynamics of
exciton-impurity scattering. To illustrate this, we set V = V0f(t) in Hamiltonian (6), and
solve the corresponding time-dependent Schrodinger equation. We expand the eigenstate
of the system in the free-exciton basis, |#(t)" ="
k C(k, t)|#k"e!iE(k)t/!, and integrate the
corresponding 2Nmol % 2Nmol system of first-order di"erential equations for the complex
amplitudes C(k, t).
6
Off-diagonal disorder!
Diagonal disorder!
-500 -250 0 250 500x (a)
0
10
20
30
40
50
60|Ψ
(x)|2 (
1/N
mol
)
-500 -250 0 250 500x (a)
0
2
4
6
|Ψ(x
)|2 (1/
Nm
ol)
-500 -250 0 250 500x (a)
100
200
300
400No diagonal disorderStrong diagonal disorder
Diagonal disorder ~ off-diagonal disorder
Applications!• Time-domain quantum simulation of localization of quantum particles:!
timescale of Anderson localization ! dynamics of exciton localization as a function of effective mass, exciton ! bandwidth, and exciton-impurity interaction strength ! effect of disorder correlations on localization and delocalization!
• Negative refraction of MW fields!
• Controlled preparation of many-body entangled states of molecules!
• Effects of dimensionality and finite size on energy transfer in crystals!
Energy diagram of a 2Σ diatomic molecule
How do electric fields affect spin relaxation?
• Induce couplings between the rotational levels (!N = 1)
• Increase the energy gap between the rotational levels
R. V. Krems, A.Dalgarno, N.Balakrishnan, and G.C. Groenenboom, PRA 67, 060703(R) (2003)
0 200 400 600 800 1000B(mT)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Ene
rgy
(cm
-1)
0 5 10 15 20B(mT)
14800
15000
15200
15400E
nerg
y (M
Hz)
534 536 538 540B(mT)
7400
7425
7450
7475
7500
7525
Ene
rgy
(MH
z)
α
βγ
γ
γ
β
530 540 550 560 570 580-5
-4
-3
-2
-1
0C
oupl
ing
Ene
rgy
(kH
z)
E=1 kV/cmE=2 kV/cmE=5 kV/cm
530 540 550 560 570 580B (mT)
0
5
10
15
20
Exc
iton
Ban
dwid
th (
kHz)
-1000 -500 0 500 10000
1
2
3
|Ψ|2 (
1/N
mol
)
-1000 -500 0 500 10000
1
2
3
4
-1000 -500 0 500 1000x (in units of a)
0
5
10
15
20
|Ψ|2 (
1/N
mol
)
-1000 -500 0 500 1000x (in units of a)
0
10
20
30
40
(c) (d)
(a) (b)
Frenkel exciton
φn = |00〉1|00〉2...|10〉n|00〉n+1...|00〉N
ψk =∑n
eik·rn√Nφn
Frenkel exciton
φn = |00〉1|00〉2...|10〉n|00〉n+1...|00〉N
ψk =∑n
eik·rn√Nφn
Ψ =1√Nmol
∑i
CiΦSi
ΦSi = |MS = 1/2〉ri∏j 6=i|MS = −1/2〉rj.
Frenkel exciton
φn = |00〉1|00〉2...|10〉n|00〉n+1...|00〉N
ψk =∑n
eik·rn√Nφn
Ψ =1√Nmol
∑i
CiΦSi
ΦSi = |MS = 1/2〉ri∏j 6=i|MS = −1/2〉rj.
α| ↑〉| ↓〉 + β| ↓〉| ↑〉
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1B = 536.9 mT
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1B = 536.8 mT
0 1 2 3 4 5t (ms)
0
0.2
0.4
0.6
0.8
1B = 536.7 mT
0 1 2 3 4 5t (ms)
0
0.2
0.4
0.6
0.8
1B = 536.5 mT
|A(t
)|2
Applications!
• Time-domain quantum simulation of localization of quantum particles:!
timescale of Anderson localization ! dynamics of exciton localization as a function of effective mass, exciton ! bandwidth, and exciton-impurity interaction strength ! effect of disorder correlations on localization and delocalization!
• Negative refraction of MW fields!
• Controlled preparation of many-body entangled states of molecules!
• Effects of dimensionality and finite size on energy transfer in crystals!
Crystal with tunable impurities:!
Optical lattice of magnetic molecules:!• Crystal with tunable magnetic properties, tunable spin waves!
• Preparation of many-body entangled states of spin up-down pairs!
• ???!
Tunable exciton – phonon interactions = �Tunable Holstein Hamiltonian �
H = Hex + Hph + HI
Hex =∑i
(εeg +Dij
)B†i Bi +
∑i,j 6=i
Ji,jB†i Bj
Hph = ~ω0∑q,λ
(a†q,λaq,λ +
1
2)
HI =1
2
∑i,j 6=i
(a†i + ai − a
†j − aj
)×{
gDij
[B†i Bi + B
†jBj
]+ gJij
[B†i Bj + B
†jBi
]}
0 2 4 6 8 10Electric field (kV/cm)
-30
-20
-10
0
10
20
30
Ene
rgy
(kH
z)
0 2 4 6 8 10Electric field (kV/cm)
-40
-20
0
20
40
60gJ
gJ
gD
gD
One dimensional array of LiCs molecules
Θ = 90 Θ = 0
0
0.2
0.4
0.6
0.8
1Pr
obab
ility
0 100 200 300 400Time (µs)
0
0.2
0.4
0.6
0.8
1
Prob
abili
ty
0 100 200 300 400t (µs)
02468
10
E (
kV/c
m)
No phonons
With phonons
Quantum particles with tunable quantum statistics…
Kinematic Interaction
One-particle state:
Two-particle state:
Bose/Fermi
other
Operators are neither bosonic, nor fermionic
What does this mean?
The same molecule cannot be excited twice!
The two excitations are coupled!
k1 and k2 are not conserved, however the total wavevector K = k1 + k2 should be conserved
Two-Particle Schroedinger Equation
In terms of K and k:
Bound state solutions?
Notation:
No solutions …
However,
Schroedinger equation has a simple solution
if the dispersion curve shape is such that for a specific value of the total wave vector K=K* (and specific branches ρ1 and ρ2) the sum does not depend on the relative wave vector k. This solution is (N-1)-time degenerate (n-degeneracy) and has the following wave function:
n is the (fixed) distance between excitations in site representation!
Under this condition
Can happen for branches with high symmetry:
K
|k|=0 |k|=π/4a
|k|=3π/4a |k|=π/a
2π/a -2π/a
Can be realized with Frenkel rotational excitons in an optical lattice with oblique electric field:
E θ
E = 10 kV/cm E = 0
What can we do with molecules on a lattice that cannot be done with conventional crystals?
1. Study rotational excitons: Rotational excitons are controllable * Electric field can be used to control exciton effective mass, exciton – impurity interactions and exciton – exciton interactions
2. Study quantum statistics of excitons * The role of kinematic interactions remains an open question 3. Study energy transfer in molecular ensembles * Could be used for quantum simulation of energy transfer in photosynthetic complexes and polaron physics
ReferencesFelipe Herrera, Marina Litinskaya, and RK,space holder space Phys. Rev. A 82, 033428 (2010).
Jesus Perez-Rios, Felipe Herrera and RK,space holder space New J. Phys. 12, 103007 (2010).
Felipe Herrera and RK, arXiv:1010.1782
T. V. Tscherbul and RK, PRL 97, 083201 (2006).
Related ReviewsR. V. Krems, Perspective on “Cold Controlled Chemistry”,fill this space Phys. Chem. Chem. Phys. 10, 479 (2008).
R. V. Krems, Int. Rev. Phys. Chem. 24, 99 (2005).
Book
R. V. Krems, W. C. Stwalley, and B. Friedrich (eds.), “Cold Molecules:Theory, Experiment, Applications”, CRC Press (2009) - 750 pages.
