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CONFINED
FEW-BODY SYSTEMS
Alejandro Saenz
AG Moderne Optik
Institut fur Physik
Humboldt-Universitat zu Berlin
(614. Heraeus-Seminar, Bad Honnef, 20.04.2016)
Theory group “Modern Optics” at HU Berlin
PhD students
Johann Forster
Alvaro Magana
Khaled Mohamed Almhdi
Eric Ouma Jobunga
Stephen Onyango Okeyo
Christoph Roll
Bruno Schulz
Students (Master/Bachelor)
Luisa Esguerra
Maja-Olivia Lenz
Guohua Maier
Marty Oelschlager
Maike Ostmann
Bettina Beverungen
Eduard Dronnik
Manuel Schneider
Ingmar Schubert
Tobias Uihlein
Financial support:
A. Saenz: Confined few-body systems (2) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Overview
• Interactions in ultracold gases.
A. Saenz: Confined few-body systems (3) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Overview
• Interactions in ultracold gases.
• Influence of confinement (trap or optical lattice).
A. Saenz: Confined few-body systems (3) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Overview
• Interactions in ultracold gases.
• Influence of confinement (trap or optical lattice).
• Consequences of reduced dimensionality.
A. Saenz: Confined few-body systems (3) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Overview
• Interactions in ultracold gases.
• Influence of confinement (trap or optical lattice).
• Consequences of reduced dimensionality.
• Confinement-induced resonances:
? as introduced by Olshanii
A. Saenz: Confined few-body systems (3) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Overview
• Interactions in ultracold gases.
• Influence of confinement (trap or optical lattice).
• Consequences of reduced dimensionality.
• Confinement-induced resonances:
? as introduced by Olshanii
? experimental detection: the puzzle
A. Saenz: Confined few-body systems (3) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Overview
• Interactions in ultracold gases.
• Influence of confinement (trap or optical lattice).
• Consequences of reduced dimensionality.
• Confinement-induced resonances:
? as introduced by Olshanii
? experimental detection: the puzzle
? solution of the puzzle.
A. Saenz: Confined few-body systems (3) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Overview
• Interactions in ultracold gases.
• Influence of confinement (trap or optical lattice).
• Consequences of reduced dimensionality.
• Confinement-induced resonances:
? as introduced by Olshanii
? experimental detection: the puzzle
? solution of the puzzle.
• Summary.
A. Saenz: Confined few-body systems (3) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Simplified atom–atom interaction model
Concept (cf. nuclear or solid-state physics):
• In many cases the wavefunction of a system consists of a complicated short-rangeand a “simple” long-range part.
A. Saenz: Confined few-body systems (4) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Simplified atom–atom interaction model
Concept (cf. nuclear or solid-state physics):
• In many cases the wavefunction of a system consists of a complicated short-rangeand a “simple” long-range part.
• Sometimes, the physics is “only” determined by the long-range part
(for example valence electrons for (metallic) conduction).
A. Saenz: Confined few-body systems (4) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Simplified atom–atom interaction model
Concept (cf. nuclear or solid-state physics):
• In many cases the wavefunction of a system consists of a complicated short-rangeand a “simple” long-range part.
• Sometimes, the physics is “only” determined by the long-range part
(for example valence electrons for (metallic) conduction).
• The short-range of the potential influences, however, the long-range solution(phase shift).
A. Saenz: Confined few-body systems (4) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Simplified atom–atom interaction model
Concept (cf. nuclear or solid-state physics):
• In many cases the wavefunction of a system consists of a complicated short-rangeand a “simple” long-range part.
• Sometimes, the physics is “only” determined by the long-range part
(for example valence electrons for (metallic) conduction).
• The short-range of the potential influences, however, the long-range solution(phase shift).
• Substitute the correct potential by a pseudopotential that yields the samewavefunction in the outer regime.
A. Saenz: Confined few-body systems (4) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Simplified atom–atom interaction model
Concept (cf. nuclear or solid-state physics):
• In many cases the wavefunction of a system consists of a complicated short-rangeand a “simple” long-range part.
• Sometimes, the physics is “only” determined by the long-range part
(for example valence electrons for (metallic) conduction).
• The short-range of the potential influences, however, the long-range solution(phase shift).
• Substitute the correct potential by a pseudopotential that yields the samewavefunction in the outer regime.
Atom-atom interaction: Vmol(R) → Vpseudo(R) = 4π ~2
µR2 asc δ(R)
A. Saenz: Confined few-body systems (4) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Simplified atom–atom interaction model
Concept (cf. nuclear or solid-state physics):
• In many cases the wavefunction of a system consists of a complicated short-rangeand a “simple” long-range part.
• Sometimes, the physics is “only” determined by the long-range part
(for example valence electrons for (metallic) conduction).
• The short-range of the potential influences, however, the long-range solution(phase shift).
• Substitute the correct potential by a pseudopotential that yields the samewavefunction in the outer regime.
Atom-atom interaction: Vmol(R) → Vpseudo(R) = 4π ~2
µR2 asc δ(R)
Note: Vpseudo is counterintuitive: long-range behaviour described by δ function!!!
A. Saenz: Confined few-body systems (4) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Tunable interaction: magnetic Feshbach resonances
∆(B)
E
R
Simple picture:
Only 2 channels:
− open (continuum) channel,
− closed (bound) channel.
A. Saenz: Confined few-body systems (5) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Tunable interaction: magnetic Feshbach resonances
∆(B)
E
R 50 100 150 200R (units of a0)
-4
-3
-2
-1
0
1
2
Cha
nnel
fun
ctio
ns ψ
χ(R
)
5 10 20 30-1
-0.5
0
0.5
1
Simple picture:
Only 2 channels:
− open (continuum) channel,
− closed (bound) channel.
Multichannel reality:
Example 6Li-87Rb : 8 coupled channels,
− very different length scales involved,
− high quality molecular potential curves
required.
A. Saenz: Confined few-body systems (5) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Tuning the interparticle interaction
Magnetic Feshbach resonance: magnetic field modifies scattering length a.
Scattering length determines interparticle interaction.
−→ Tuning the interparticle interaction with a magnetic field!
A. Saenz: Confined few-body systems (6) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Intermezzo: SFA (KFR) approximation
Sfi = limt→∞〈Φf |Ψ(+)
i 〉 = limt→−∞
〈Ψ(−)f |Φi〉
A. Saenz: Confined few-body systems (7) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Intermezzo: SFA (KFR) approximation
Sfi = limt→∞〈Φf |Ψ(+)
i 〉 = limt→−∞
〈Ψ(−)f |Φi〉
Keldysh (JETP 20, 1307 (1965)):
• Final state: ignore influence of remaining ion on ionized electron.−→ Electron described by Volkov wavefunction.
• Problem: derived for short-range potential → Coulomb-corrections?
A. Saenz: Confined few-body systems (7) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Intermezzo: SFA (KFR) approximation
Sfi = limt→∞〈Φf |Ψ(+)
i 〉 = limt→−∞
〈Ψ(−)f |Φi〉
Keldysh (JETP 20, 1307 (1965)):
• Final state: ignore influence of remaining ion on ionized electron.−→ Electron described by Volkov wavefunction.
• Problem: derived for short-range potential → Coulomb-corrections?
Alternative formulations (quite confusing):
• Keldysh: length gauge (L), reversed-time S matrix.
• Faisal: velocity gauge (V), direct-time S matrix.
• Reiss: velocity gauge (V), reversed-time S matrix.
A. Saenz: Confined few-body systems (7) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Intermezzo: SFA (KFR) approximation
Sfi = limt→∞〈Φf |Ψ(+)
i 〉 = limt→−∞
〈Ψ(−)f |Φi〉
Keldysh (JETP 20, 1307 (1965)):
• Final state: ignore influence of remaining ion on ionized electron.−→ Electron described by Volkov wavefunction.
• Problem: derived for short-range potential → Coulomb-corrections?
Alternative formulations (quite confusing):
• Keldysh: length gauge (L), reversed-time S matrix.
• Faisal: velocity gauge (V), direct-time S matrix.
• Reiss: velocity gauge (V), reversed-time S matrix.
• W. Becker: dressing of initial state required.
• Mishima et al., Chao: “exact” Keldysh theory:use residue theorem instead of saddle-point approximation.
A. Saenz: Confined few-body systems (7) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Validity of the strong-field approximation (SFA): 800 nm
0 1 2 3 4 5 6 7 8 9 10
Intensity, 1013
W/cm2
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
Ion
izat
ion
rat
e, a
.u.
Floquet
TDSE
K-SFA
B-SFA
L-SFA
V-SFA
Ionization rates of H atoms (λ=800 nm)
L - length gauge, V - velocity gauge, K - Krainov corrected L-SFA, B - Becker corrected V-SFA.
A. Saenz: Confined few-body systems (8) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Validity of the strong-field approximation (SFA): 400 nm
0 1 2 3 4 5 6 7 8 9 10
Intensity, 1013
W/cm2
10-7
10-6
10-5
10-4
10-3
Ioniz
atio
n r
ate,
a.u
.Floquet
TDSE
K-SFA
B-SFA
L-SFA
V-SFA
Ionization rates of H atoms (λ=400 nm)
L - length gauge, V - velocity gauge, K - Krainov corrected L-SFA, B - Becker corrected V-SFA.
A. Saenz: Confined few-body systems (9) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Optical lattices: shaped (tight) confinement
Counterpropagating lasers:
−→ standing light field.
Trap potential varies as
Ulat sin2(~k~r )
with
k = 2πλ
λ: laser wavelength.
Ulat ∝ I α(λ)
with
laser intensity I and
atomic polarizability α.
[reproduced from I. Bloch, Nature Physics 1, 23 (2005)]
A. Saenz: Confined few-body systems (10) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Why is few-body physics of interest?
Mott state with 1 or 2 atoms/molecules:
• One atom per site: isolated addressable quantum system:
interesting for quantum information (qubit register).
A. Saenz: Confined few-body systems (11) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Why is few-body physics of interest?
Mott state with 1 or 2 atoms/molecules:
• One atom per site: isolated addressable quantum system:
interesting for quantum information (qubit register).
• Protection against (unwanted) collisions.
A. Saenz: Confined few-body systems (11) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Why is few-body physics of interest?
Mott state with 1 or 2 atoms/molecules:
• One atom per site: isolated addressable quantum system:
interesting for quantum information (qubit register).
• Protection against (unwanted) collisions.
• Two atoms or molecules per site: controlled interactions
(quantum-state resolved reactions: “reaction chamber”)
A. Saenz: Confined few-body systems (11) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Why is few-body physics of interest?
Mott state with 1 or 2 atoms/molecules:
• One atom per site: isolated addressable quantum system:
interesting for quantum information (qubit register).
• Protection against (unwanted) collisions.
• Two atoms or molecules per site: controlled interactions
(quantum-state resolved reactions: “reaction chamber”)
• Molecules: heteronuclear diatomics possess electric dipole moments:
orientational dependent interaction (cf. Ising model).
A. Saenz: Confined few-body systems (11) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Why is few-body physics of interest?
Mott state with 1 or 2 atoms/molecules:
• One atom per site: isolated addressable quantum system:
interesting for quantum information (qubit register).
• Protection against (unwanted) collisions.
• Two atoms or molecules per site: controlled interactions
(quantum-state resolved reactions: “reaction chamber”)
• Molecules: heteronuclear diatomics possess electric dipole moments:
orientational dependent interaction (cf. Ising model).
Required: Full understanding of few-body systems in optical lattices
(static and dynamic properties).
A. Saenz: Confined few-body systems (11) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Pseudopotential approximation in traps
Atom-atom interaction: Vmol(R) → Vpseudo(R) = 4π ~2
µR2 asc δ(R)
• This relation was derived for k → 0 (limit of zero-collision energy).
• In a (tight) trap energy is quantized: zero-point motion.
• Intercept of Ψ on R axis does not agree with asc.
Deviation for Ψ small, e. g., 10 kHz trap yields intercept on
−2023 for asc = −2030 a0.
This is not true for Ψpseudo: intercept on −1447 for asc = −2030 a0.
−→ Wrong prediction of photoassociation window (for asc 0).
• Introduce an energy-dependent asc(E) that inserted in Vpseudo(R) matches
(for E = 32 ~ωtrap) the correct Ψ (at R →∞).
Note: In contrast to the physical asc the empirical parameter asc(E) follows onlyfrom the correct Ψ obtained with Vmol(R)!
A. Saenz: Confined few-body systems (12) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Validity of the pseudopotential approximation
0 20000 40000Interatomic separation R (a.u.)
0
0.002
0.004
0.006
0.008W
avef
unct
ion
(a.u
.-1/2
)
50 100 150 200Interatomic separation R (a.u.)
0
0.001
0.002
0.003
Wav
efun
ctio
n (a
.u.-1
/2)
Spin-polarized 6Li atoms (a 3Σu) in a 10 kHz trap:
“correct” wavefunction (black, asc = −2030 a0) vs. energy independent (red, asc = −2030 a0)
and dependent (blue, asc = −2872 a0) pseudopotential results.
A. Saenz: Confined few-body systems (13) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Theoretical challenges:
• Non-trivial, non-analytic atom-atom interaction (unlike Coulomb interaction).
• Magnetic Feshbach resonances: multi-scale, multi-channel problem.
Multi-channel R-matrix approach (incl. combined exp. and theor. determinationof 7Li87Rb resonances) [Phys. Rev. A 79, 012717 (2009)].
A. Saenz: Confined few-body systems (14) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Theoretical challenges:
• Non-trivial, non-analytic atom-atom interaction (unlike Coulomb interaction).
• Magnetic Feshbach resonances: multi-scale, multi-channel problem.
Multi-channel R-matrix approach (incl. combined exp. and theor. determinationof 7Li87Rb resonances) [Phys. Rev. A 79, 012717 (2009)].
• Theory for magnetic Feshbach resonances derived for free space.
A. Saenz: Confined few-body systems (14) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Theoretical challenges:
• Non-trivial, non-analytic atom-atom interaction (unlike Coulomb interaction).
• Magnetic Feshbach resonances: multi-scale, multi-channel problem.
Multi-channel R-matrix approach (incl. combined exp. and theor. determinationof 7Li87Rb resonances) [Phys. Rev. A 79, 012717 (2009)].
• Theory for magnetic Feshbach resonances derived for free space.
Influence of lattice (confinement) on magnetic Feshbach resonances?
A. Saenz: Confined few-body systems (14) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Magnetic Feshbach resonances (MFRs) in a harmonic trap
• Description as coupled single open and closed channels (|Ψ〉 = C|open〉+ A|closed〉)• Use analytically known long-range behavior of the wave functions (parabolic cylinder fcts.)
A. Saenz: Confined few-body systems (15) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Magnetic Feshbach resonances (MFRs) in a harmonic trap
• Description as coupled single open and closed channels (|Ψ〉 = C|open〉+ A|closed〉)• Use analytically known long-range behavior of the wave functions (parabolic cylinder fcts.)
With this one can
A. Saenz: Confined few-body systems (15) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Magnetic Feshbach resonances (MFRs) in a harmonic trap
• Description as coupled single open and closed channels (|Ψ〉 = C|open〉+ A|closed〉)• Use analytically known long-range behavior of the wave functions (parabolic cylinder fcts.)
With this one can
1. recover the known energy relation in the trap
(aho =√~/mω)
a
aho
= f(E) ≡Γ (1/4− E/2~ω)
Γ (3/4− E/2~ω)
A. Saenz: Confined few-body systems (15) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Magnetic Feshbach resonances (MFRs) in a harmonic trap
• Description as coupled single open and closed channels (|Ψ〉 = C|open〉+ A|closed〉)• Use analytically known long-range behavior of the wave functions (parabolic cylinder fcts.)
With this one can
1. recover the known energy relation in the trap
(aho =√~/mω)
a
aho
= f(E) ≡Γ (1/4− E/2~ω)
Γ (3/4− E/2~ω)
2. derive the energy-dependent scattering length
a(E,B) = abg
(1−
∆B
B − B0 + δB − E/µ
)
in contrast to a previously suggested form
a(E,B) = abg
(1−
∆B(1 + (kabg)
2)
B − B0 + δB + (kabg)2∆B − E/µ
)
A. Saenz: Confined few-body systems (15) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Magnetic Feshbach resonances (MFRs) in a harmonic trap
• Description as coupled single open and closed channels (|Ψ〉 = C|open〉+ A|closed〉)• Use analytically known long-range behavior of the wave functions (parabolic cylinder fcts.)
With this one can
1. recover the known energy relation in the trap
(aho =√~/mω)
a
aho
= f(E) ≡Γ (1/4− E/2~ω)
Γ (3/4− E/2~ω)
2. derive the energy-dependent scattering length
a(E,B) = abg
(1−
∆B
B − B0 + δB − E/µ
)
in contrast to a previously suggested form
a(E,B) = abg
(1−
∆B(1 + (kabg)
2)
B − B0 + δB + (kabg)2∆B − E/µ
)
-2 0 2 4 6
-4
-2
0
2
4
EÑΩ
fHE
L,aH
EL
a ho
aaho = f HELf HELaaho
(Shift δB and slope
µ = ERBS(B)/(B−B0)
exp. predictable.)
A. Saenz: Confined few-body systems (15) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Magnetic Feshbach resonances (MFRs) in a harmonic trap
• Description as coupled single open and closed channels (|Ψ〉 = C|open〉+ A|closed〉)• Use analytically known long-range behavior of the wave functions (parabolic cylinder fcts.)
With this one can
1. recover the known energy relation in the trap
(aho =√~/mω)
a
aho
= f(E) ≡Γ (1/4− E/2~ω)
Γ (3/4− E/2~ω)
2. derive the energy-dependent scattering length
a(E,B) = abg
(1−
∆B
B − B0 + δB − E/µ
)
in contrast to a previously suggested form
a(E,B) = abg
(1−
∆B(1 + (kabg)
2)
B − B0 + δB + (kabg)2∆B − E/µ
)
-2 0 2 4 6
-4
-2
0
2
4
EÑΩ
fHE
L,aH
EL
a ho
aaho = f HELf HELaaho
(Shift δB and slope
µ = ERBS(B)/(B−B0)
exp. predictable.)
a
b
c
d
A. Saenz: Confined few-body systems (15) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Magnetic Feshbach resonances (MFRs) in a harmonic trap
• Description as coupled single open and closed channels (|Ψ〉 = C|open〉+ A|closed〉)• Use analytically known long-range behavior of the wave functions (parabolic cylinder fcts.)
With this one can
1. recover the known energy relation in the trap
(aho =√~/mω)
a
aho
= f(E) ≡Γ (1/4− E/2~ω)
Γ (3/4− E/2~ω)
2. derive the energy-dependent scattering length
a(E,B) = abg
(1−
∆B
B − B0 + δB − E/µ
)
in contrast to a previously suggested form
a(E,B) = abg
(1−
∆B(1 + (kabg)
2)
B − B0 + δB + (kabg)2∆B − E/µ
)
-2 0 2 4 6
-4
-2
0
2
4
EÑΩ
fHE
L,aH
EL
a ho
aaho = f HELf HELaaho
(Shift δB and slope
µ = ERBS(B)/(B−B0)
exp. predictable.)
a
b
c
d
3. derive the admixture of the closed channel
A
C∝f(E)− abg/aho√
f ′(E)
A. Saenz: Confined few-body systems (15) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
How good is the model?
Comparison with full coupled-channel calculations for 6Li-87Rb in a 200 kHz trap:
1065 1066 1067 1068
-2
0
2
4
6
B Hunits of GaussL
EHu
nits
ofÑΩL
1065 1066 1067 1068 10690.001
0.01
0.1
1.
B Hunits of GaussLR
BS
cont
ribu
tion
A2
MC result
1st energy level
2nd energy level
3rd energy level
4th energy level
• Energy deviation < 0.003 ~ω.
• Closed-channel admixture deviation < 0.1%.
A. Saenz: Confined few-body systems (16) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Explaining a long-standing discrepancy
• Resonances of a ∝ f(E) are located at E(n)res = ~ω(2n+ 1
2)⇒ thus NOT at bare resonance
position BR = B0 − δB, but atB = B(n)
res = B0 − δB + E(n)res /µ .
• This explains the disagreement of experimentally observed MFR positions of 87Rb;
predicted shift of 0.034 Gauss in good agreement with experimental results.
weak dipole trap, M. Erhard et al.
Phys. Rev. A 69 032705 (2004)
tight optical trap, A. Widera et al.
Phys. Rev. Lett. 92 160406 (2004).
A. Saenz: Confined few-body systems (17) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Harmonic vs. anharmonic confinement (optical lattice)
Analytical separable solution exists for the atom pair, if
• the interatomic interaction is described by a pseudo potential(Vatom−atom ∝ asc δ(~r ) with s-wave scattering length asc),
• the harmonic approximation is adopted for the lattice potential, and
• both atoms “feel” the same lattice potential.
A. Saenz: Confined few-body systems (18) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Harmonic vs. anharmonic confinement (optical lattice)
Analytical separable solution exists for the atom pair, if
• the interatomic interaction is described by a pseudo potential(Vatom−atom ∝ asc δ(~r ) with s-wave scattering length asc),
• the harmonic approximation is adopted for the lattice potential, and
• both atoms “feel” the same lattice potential.
However, coupling of center-of-mass (COM) and relative (REL) motion
• for the (correct) sin2 potential (or any physical trap),
• even in harmonic traps, if the two atoms experience different trap potentials
? heteronuclear atom pairs or
? atoms in different electronic states.
A. Saenz: Confined few-body systems (18) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Present theoretical approach
Hamiltonian (6D):
H(~R,~r ) = hCOM(~R ) + hREL(~r ) + W(~R,~r )
with ~R : center-of-mass (COM) ~r : relative motion (REL) coordinate .
• Taylor expansion of the sin2 lattice potential (to arbitrary order).
• Also cos2, mixed, and fully anisotropic lattices possible.
• All separable terms included in either hCOM or hREL.
• Full interatomic interaction potential (typically a numerical BO curve).
• Configuration interaction (CI) type full solution using the eigenfunctions
(orbitals) of hCOM and hREL.
• Full consideration of lattice symmetry (and possible indistinguishability of atoms).
• Arbitrary isotropic (or additional anisotropic dipole-dipole) interaction.
A. Saenz: Confined few-body systems (19) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Heteronuclear-molecule formation in optical lattices
5 10 15 200−50
2
4
6
8
10
atom(mole ule)number/104
E· h−1/kHz
|9/2,−9/2〉
Eb
|9/2,−9/2〉
Mole ules
EEb
|9/2,−7/2〉
rf
544.5 545.0 545.5 546.0 546.5 547.0 547.5 548.0 548.5-140-120-100-80-60-40-2002040
B / G
E·h
−1
/kHz
a < 0a > 0
onnement-indu edmole ulesrealmole ulesrepulsively inter-a ting pairs
Ulat = 27.5Er
Ulat = 40.0Er
C. Ospelkaus et al. [Phys. Rev. Lett. 97 120402 (2006)]:
• Formation of 87Rb-40K by radio frequency (rf) association.
• Measurement of the binding energy (on absolute scale).
Claim: Deviations from separable harmonic approximation are seen!
A. Saenz: Confined few-body systems (20) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Two atoms in a single well: anharmonicity and coupling
We obtained exact solutions
for two interacting atoms in
one well of an OL.
A. Saenz: Confined few-body systems (21) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Two atoms in a single well: anharmonicity and coupling
We obtained exact solutions
for two interacting atoms in
one well of an OL.
Agreement with experiment on kHz level→ improved resonance parameters by fit?
Fit works only, if anharmonicity is considered
→ coupling of COM and REL motion important!
544 544.5 545 545.5 546 546.5 547 547.5 548B (G)
-120
-90
-60
-30
0
30
Ene
rgy
( h
kHz
)∆B= 3G, B0
fit=546.66G
∆B= 3G, B0ex=546.8G
ascE with ∆B, B0
ex
expt. data"real"
asc > 0
confinement-induced
molecule
repulsivelyinteracting pair
molecule
asc < 0
∆B
B0ex
[S. Grishkevich et al., Phys. Rev. A 80, 013403 (2009)]
A. Saenz: Confined few-body systems (21) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Few-body physics for improving many-body models
We obtain exact solutions for
two interacting atoms in 3 wells
of an OL.
• Comparison with BH model with Hamiltonian
HBH = J∑
<i,j>
b†i bj +
U
2
∑
i
ni(ni − 1) +∑
i
εib†i bi
yields optimal BH parameters Jopt, Uopt, εopti
and validity range of BH model.
0x
Energy
Ε-1 Ε0 Ε1
J
U
V0
-Πk0 Πk0
A. Saenz: Confined few-body systems (22) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Few-body physics for improving many-body models
We obtained exact solutions
for two interacting atoms in 3
wells of an OL.
• Introduction of improved U parameter by correction
of harmonic interaction energy: U corr = AUharm with
A = 2
(π~mω
)32∫d
3~r |w0(~r )|4
0.5 1.0 1.5 2.0-6
-5
-4
-3
-2
-1
01 5 10 15
V0 @ÑΩD
U@E
rD
V0 @ErD
Uopt
UBH
Uharm
Ucorr
asc= -4638 a0
more in Phys. Rev. A 80 013404 (2009)
A. Saenz: Confined few-body systems (23) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Reduced dimension: fermionization of bosons (1D vs. quasi 1D)
1.0×104
2.0×104
3.0×104
4.0×104
5.0×104
6.0×104
r [a.u.]
0
1e-05
2e-05
3e-05
4e-05
5e-05
6e-05
ρ(r
)
dz/a
0=2.6, η= 3.2
dz/a
0=3.1, η= 4.5
dz/a
0=4.6, η=10.0
Radial density of two atoms in a quasi-1D (cigar-shaped) confinement:
− scattering length a0 = 5624 a.u.
− anisotropy η = (dz/d⊥)2
− transversal trap length d⊥ = 1.46 a0
− full Born-Oppenheimer potential.
A. Saenz: Confined few-body systems (24) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Reduced dimension: fermionization of bosons (1D vs. quasi 1D)
1.0×104
2.0×104
3.0×104
4.0×104
5.0×104
6.0×104
r [a.u.]
0
1e-05
2e-05
3e-05
4e-05
5e-05
6e-05
ρ(r
)
dz/a
0=2.6, η= 3.2
dz/a
0=3.1, η= 4.5
dz/a
0=4.6, η=10.0
1.0×104
2.0×104
3.0×104
4.0×104
5.0×104
6.0×104
r [a.u.]
0
1e-05
2e-05
3e-05
4e-05
5e-05
6e-05
ρ(r
)
dz/a
0=2.6, η= 3.2
dz/a
0=3.1, η= 4.5
dz/a
0=4.6, η=10.0
Fermions, η=10.0
Radial density of two atoms in a quasi-1D (cigar-shaped) confinement:
− scattering length a0 = 5624 a.u.
− anisotropy η = (dz/d⊥)2
− transversal trap length d⊥ = 1.46 a0
− full Born-Oppenheimer potential.
A. Saenz: Confined few-body systems (24) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Confinement-induced resonances (CIR)
Relative-motion s-wave scattering theory for two ultracold atoms in an harmonicquasi 1D confinement: mapping of quasi-1D system onto pure 1D system.
Renormalized 1D interaction strength [M. Olshanii, PRL 81, 938 (1998)]:
g1D =2a~2
µd2⊥
1
1 + ζ(12) a
d⊥
a := s-wave scattering length d⊥ =√
~µω⊥
: transversal confinement
µ := reduced mass ζ(x) =∑∞k=1 k
−x
Resonance: g1D →∞ for d⊥a = −ζ(1
2) ≈ 1.46 . . .
Analogously: confinement-inuced resonance occurs also in (quasi) 2D
[Petrov, Holzmann, Shlyapnikov, PRL 84, 2551 (2000)].
A. Saenz: Confined few-body systems (25) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Olshanii’s model (I)
Resonance occurs if artificially excited bound state crosses the free ground-statethreshold:
-5 0 5
d⊥ /a
0.5
1
1.5
2
2.5
3
3.5
E [
h_ ω⊥]
CIRd⊥ /a=-ζ(1/2)
Blue: quasi 1D spectrum
Red: artificially(!) ex-cited bound state
Green: quasi continuumthreshold
A. Saenz: Confined few-body systems (26) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Olshanii’s model (II)
T. Bergeman et al., PRL 91, 163201 (2003)
Result:
Confinement-induced resonances (CIR) are not an artefact of the δ potential.
Note: No data points on shifted state!
A. Saenz: Confined few-body systems (27) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Innsbruck experiment (Cs atoms)
Blue curve: Atom losses for ωx = ωy ωz (anisotropy fixed, a varied).
Red and blue curves: Atom losses for ωx 6= ωy ωzE. Haller et al., PRL 104, 153203 (2010)
A. Saenz: Confined few-body systems (28) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Problem: agreement and conflict with theory
1 1.05 1.1 1.15 1.2 1.25ω
x / ω
y
0.5
0.6
0.7
0.8
aCIR
/ d y
RHM modelExperiment minima
E. Haller et al., PRL, 104, 153203 (2010)
⇒ Good agreement with Olshanii prediction for single anisotropy (ωx = ωy)
A. Saenz: Confined few-body systems (29) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Problem: agreement and conflict with theory
1 1.05 1.1 1.15 1.2 1.25ω
x / ω
y
0.5
0.6
0.7
0.8
aCIR
/ d y
RHM modelExperiment minima
E. Haller et al., PRL, 104, 153203 (2010)
⇒ Good agreement with Olshanii prediction for single anisotropy (ωx = ωy)
⇒ Olshanii theory: no splitting (ωx 6= ωy)!!! Peng et al., PRA 82, 063633 (2010)
A. Saenz: Confined few-body systems (29) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Complete confusion:
Innsbruck loss experiment (Haller et al.):
• Position of 1D CIR agrees with Olshanii prediction for ωx = ωy.
A. Saenz: Confined few-body systems (30) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Complete confusion:
Innsbruck loss experiment (Haller et al.):
• Position of 1D CIR agrees with Olshanii prediction for ωx = ωy.
• Splitting of 1D CIR for ωx 6= ωy seems trivial, but conflicts with Olshaniitheory.
A. Saenz: Confined few-body systems (30) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Complete confusion:
Innsbruck loss experiment (Haller et al.):
• Position of 1D CIR agrees with Olshanii prediction for ωx = ωy.
• Splitting of 1D CIR for ωx 6= ωy seems trivial, but conflicts with Olshaniitheory.
• Quasi-2D: CIR appears for a with “wrong” sign compared to Petrov, Holzmann,Shlyapnikov prediction.
A. Saenz: Confined few-body systems (30) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Complete confusion:
Innsbruck loss experiment (Haller et al.):
• Position of 1D CIR agrees with Olshanii prediction for ωx = ωy.
• Splitting of 1D CIR for ωx 6= ωy seems trivial, but conflicts with Olshaniitheory.
• Quasi-2D: CIR appears for a with “wrong” sign compared to Petrov, Holzmann,Shlyapnikov prediction.
• Quasi-2D: No losses at the “correct” value of a.
A. Saenz: Confined few-body systems (30) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Complete confusion:
Innsbruck loss experiment (Haller et al.):
• Position of 1D CIR agrees with Olshanii prediction for ωx = ωy.
• Splitting of 1D CIR for ωx 6= ωy seems trivial, but conflicts with Olshaniitheory.
• Quasi-2D: CIR appears for a with “wrong” sign compared to Petrov, Holzmann,Shlyapnikov prediction.
• Quasi-2D: No losses at the “correct” value of a.
Cambridge radio-frequency experiment (Froehlich et al.):
• Quasi-2D: CIR appears at “correct” value of a (also seen by Chris Vale).
• Note: direct measurement of the binding energies.
A. Saenz: Confined few-body systems (30) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Full treatment of two atoms in quasi-1D trap:
Full Hamiltonian: center-of-mass (COM) and relative motion (REL) motion:
H(r,R) = TREL(r) + TCOM(R) + VREL(r) + VCOM(R) + Uint(r) +W (r,R)
Note:
Anharmonic optical-lattice potential ⇒ COM and REL coupling (W (r,R) 6= 0)!
-10 -5 0 5 10
x / dho
0
2
4
6
8
10
12
V(x
) /
h_ω
ho
Sextic trap Naegerl optical latticeHarmonic trap
-4 -2 0 2 4
x / dho
0
1
2
3
4
5
V(x
) /
h_ω
ho
A. Saenz: Confined few-body systems (31) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Energy spectra (cartoon)
Relative-motion spectrum in harmonic trap vs. full (rel + com) spectrum
Relative motion only
ψb: (molecular) bound state
ψ1: lowest-lying trap state
A. Saenz: Confined few-body systems (32) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Energy spectra (cartoon)
Relative-motion spectrum in harmonic trap vs. full (rel + com) spectrum
Relative motion only
ψb: (molecular) bound state
ψ1: lowest-lying trap state
Full spectrum
Φ(0,0,0): ground com state
Φ(2,0,0): excited com state
A. Saenz: Confined few-body systems (32) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Molecule formation due to confinement
Full spectrum Avoided crossing
Coupling of center-of-mass (com) and relative (rel) motion (W 6= 0):
−→ avoided crossing
−→ molecule formation possible!
A. Saenz: Confined few-body systems (33) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Energy spectra (ab initio results)
Relative-motion spectrum in harmonic trap vs. coupled spectrum in sextic trap
-10 -5 0 5 10
d⊥ / a
0
0.5
1
1.5
2
2.5
E [
h_ω
⊥]
REL
-2 0 2 4
d⊥ / a
-2
0
2
4
E /
h_ω
⊥REL + COM + COUPLING
Many crossings are found in the coupled model,
A. Saenz: Confined few-body systems (34) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Energy spectra (ab initio results)
Relative-motion spectrum in harmonic trap vs. coupled spectrum in sextic trap
-10 -5 0 5 10
d⊥ / a
0
0.5
1
1.5
2
2.5
E [
h_ω
⊥]
REL
-2 0 2 4
d⊥ / a
-2
0
2
4
E /
h_ω
⊥REL + COM + COUPLING
Many crossings are found in the coupled model,
but which of them lead to resonances?
A. Saenz: Confined few-body systems (34) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Approximate selection rules
Coupling matrix element:
W(n,m,k) = 〈φn(R)ψb(r) |W (r,R) |φm(R)ψk(r) 〉
W (r,R) =∑j=x,y,zWj(rj, Rj)
W(n,m,k) ≈ δnz,mz F(n,m,k)(W )
F(n,m,k)(W ) =
[δny,my〈φnx(X)|Wx(X)|φmx(X)〉〈ψb(r)|Wx(x)|ψk(r)〉
+δnx,mx〈φny(Y )|Wy(Y )|φmy(Y )〉〈ψb(r)|Wy(y)|ψk(r)〉]
REL bound state:|ψb(r)〉
REL trap state: ψk(r)
COM states: φn(R) =φnx(X)φny(Y )φnz(Z)
Ultracold: only ground trap state populated =⇒ m = k = 0.
Resonances:
Crossing of transversally COM excited REL bound state with ground (COM andREL) trap state.
A. Saenz: Confined few-body systems (35) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Avoided Crossings (I)
Only few crossings are avoided (approx. selection rules):
-2 0 2 4
d⊥ / a
-2
0
2
4
E /
h_
ω⊥
Large part of spectrum
1.15 1.2 1.25 1.3 1.35 1.4 1.45
d⊥ / a
2.071
2.072
2.073
2.074
2.075
E /
h_ω
⊥
|ψ(b)
Φ(2,0,0)
>
|ψ(b)
Φ(0,2,0)
>
|ψ(b)
Φ(0,2,2)
>
|ψ(b)
Φ(0,0,14)
>|ψ
1Φ
(0,0,0)>
Zoom-in in spectrum.
A. Saenz: Confined few-body systems (36) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Avoided Crossings (II)
Only few crossings are avoided (approx. selection rules):
1.15 1.2 1.25 1.3 1.35 1.4 1.45
d⊥ / a
2.071
2.072
2.073
2.074
2.075
E /
h_ω
⊥
|ψ(b)
Φ(2,0,0)
>
|ψ(b)
Φ(0,2,0)
>
|ψ(b)
Φ(0,2,2)
>
|ψ(b)
Φ(0,0,14)
>|ψ
1Φ
(0,0,0)>
ωx = ωy ωz
1.25 1.3 1.35 1.4 1.45
dy / a
2.1715
2.172
2.1725
2.173
2.1735
2.174
E /
h_ω
y
|ψbφ(2,0,0)>
|ψbφ(0,2,0)>
|ψ0φ(0,0,0)>
ωx 6= ωy ωz
⇒ single anisotropy (ωx = ωy ωz): degeneracy
⇒ totally anisotropic case ωx 6= ωy ωz: splitting[S. Sala, P.-I. Schneider, A.S., Phys. Rev. Lett. 109, 073201 (2012)]
A. Saenz: Confined few-body systems (37) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Comparison with Innsbruck Experiment
1 1.05 1.1 1.15 1.2 1.25ω
x / ω
y
0.65
0.7
0.75aC
IR /
d y
ExperimentRMH modelCRC model
Agreement not only for positions, but also for width.
Quantitative agreement also for quasi-2D resonance: a = 0.593 dy (exp.)vs. a = 0.595 dy (th.) [S. Sala, P.-I. Schneider, A.S., Phys. Rev. Lett. 109, 073201 (2012)]
A. Saenz: Confined few-body systems (38) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Preliminary summary
Our conclusion:
• Two types of resonances: elastic (Olshanii, Petrov et al.) and inelastic ones.
• Elastic CIR: no molecule formation, (almost) no losses (invisible in Innsbruckexperiment).
• Inelastic CIR: molecule formation, thus atom loss.
A. Saenz: Confined few-body systems (39) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Preliminary summary
Our conclusion:
• Two types of resonances: elastic (Olshanii, Petrov et al.) and inelastic ones.
• Elastic CIR: no molecule formation, (almost) no losses (invisible in Innsbruckexperiment).
• Inelastic CIR: molecule formation, thus atom loss.
• Quasi 1D: accidentally at similar positions (in fact overlapping), but widthsdiffers by about one order of magnitude (elastic: broader).
• Quasi 2D: positions differ even by sign of a.
A. Saenz: Confined few-body systems (39) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Preliminary summary
Our conclusion:
• Two types of resonances: elastic (Olshanii, Petrov et al.) and inelastic ones.
• Elastic CIR: no molecule formation, (almost) no losses (invisible in Innsbruckexperiment).
• Inelastic CIR: molecule formation, thus atom loss.
• Quasi 1D: accidentally at similar positions (in fact overlapping), but widthsdiffers by about one order of magnitude (elastic: broader).
• Quasi 2D: positions differ even by sign of a.
Note: The possibility to create molecules due to anharmonicity had earlier beensuggested: Bolda, Tiesinga, Julienne [PRA 71, 033404 (2005)]; Schneider, Grishkevich, A.S,
[Phys. Rev. A 80, 013404 (2009)]; Kestner, Duan [N. J. Phys. 12, 053016 (2010)].
A. Saenz: Confined few-body systems (39) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Preliminary summary
Our conclusion:
• Two types of resonances: elastic (Olshanii, Petrov et al.) and inelastic ones.
• Elastic CIR: no molecule formation, (almost) no losses (invisible in Innsbruckexperiment).
• Inelastic CIR: molecule formation, thus atom loss.
• Quasi 1D: accidentally at similar positions (in fact overlapping), but widthsdiffers by about one order of magnitude (elastic: broader).
• Quasi 2D: positions differ even by sign of a.
Note: The possibility to create molecules due to anharmonicity had earlier beensuggested: Bolda, Tiesinga, Julienne [PRA 71, 033404 (2005)]; Schneider, Grishkevich, A.S,
[Phys. Rev. A 80, 013404 (2009)]; Kestner, Duan [N. J. Phys. 12, 053016 (2010)].
However, not everyone (e.g. 2 out of 3 referees) was convinced!
A. Saenz: Confined few-body systems (39) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Criticism / alternative explanations:
• “There is no problem with the splitting of the 1D CIR, as its origin is evidentand explained in the experimental paper.”
(Explicit math (like done by Peng et al.) only confuses . . . )
A. Saenz: Confined few-body systems (40) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Criticism / alternative explanations:
• “There is no problem with the splitting of the 1D CIR, as its origin is evidentand explained in the experimental paper.”
(Explicit math (like done by Peng et al.) only confuses . . . )
• “Molecule formation from atoms requires three-body collisions.”
A. Saenz: Confined few-body systems (40) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Criticism / alternative explanations:
• “There is no problem with the splitting of the 1D CIR, as its origin is evidentand explained in the experimental paper.”
(Explicit math (like done by Peng et al.) only confuses . . . )
• “Molecule formation from atoms requires three-body collisions.”
• Our ab initio calculation used Li2 potential (assuming universality), not Cs2 asin Innsbruck experiment.
Note: our model does not assume any specific system.
A. Saenz: Confined few-body systems (40) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Criticism / alternative explanations:
• “There is no problem with the splitting of the 1D CIR, as its origin is evidentand explained in the experimental paper.”
(Explicit math (like done by Peng et al.) only confuses . . . )
• “Molecule formation from atoms requires three-body collisions.”
• Our ab initio calculation used Li2 potential (assuming universality), not Cs2 asin Innsbruck experiment.
Note: our model does not assume any specific system.
• Losses could be due to Cs-specific magnetic Feshbach resonances.
[Peng et al., Phys. Rev. A 82, 063633 (2010)]
A. Saenz: Confined few-body systems (40) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Criticism / alternative explanations:
• “There is no problem with the splitting of the 1D CIR, as its origin is evidentand explained in the experimental paper.”
(Explicit math (like done by Peng et al.) only confuses . . . )
• “Molecule formation from atoms requires three-body collisions.”
• Our ab initio calculation used Li2 potential (assuming universality), not Cs2 asin Innsbruck experiment.
Note: our model does not assume any specific system.
• Losses could be due to Cs-specific magnetic Feshbach resonances.
[Peng et al., Phys. Rev. A 82, 063633 (2010)]
• Multichannel CIR effect.[Melezhik, Schmelcher Phys. Rev. A 84, 042712 (2011)]
A. Saenz: Confined few-body systems (40) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Criticism / alternative explanations:
• “There is no problem with the splitting of the 1D CIR, as its origin is evidentand explained in the experimental paper.”
(Explicit math (like done by Peng et al.) only confuses . . . )
• “Molecule formation from atoms requires three-body collisions.”
• Our ab initio calculation used Li2 potential (assuming universality), not Cs2 asin Innsbruck experiment.
Note: our model does not assume any specific system.
• Losses could be due to Cs-specific magnetic Feshbach resonances.
[Peng et al., Phys. Rev. A 82, 063633 (2010)]
• Multichannel CIR effect.[Melezhik, Schmelcher Phys. Rev. A 84, 042712 (2011)]
• Losses in a many-body system (Innsbruck experiment) are very unspecific, incontrast to Cambridge rf experiment.
A. Saenz: Confined few-body systems (40) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Experimental test (with group of S. Jochim)
Exclusion of many-body and multi-channel effects:
Experiment with exactly two Li atoms in high-fidelity ground state
cf. [Serwane et al., Science 332, 336 (2011)]
A. Saenz: Confined few-body systems (41) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Experimental test (with group of S. Jochim)
Exclusion of many-body and multi-channel effects:
Experiment with exactly two Li atoms in high-fidelity ground state
cf. [Serwane et al., Science 332, 336 (2011)]
1. Confirmation of the elastic CIR by measuring the tunnel rate:
Interaction energy shifts two-atom ground state ⇒ modified atomic tunnel rate.
A. Saenz: Confined few-body systems (41) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Experimental test (with group of S. Jochim)
Exclusion of many-body and multi-channel effects:
Experiment with exactly two Li atoms in high-fidelity ground state
cf. [Serwane et al., Science 332, 336 (2011)]
1. Confirmation of the elastic CIR by measuring the tunnel rate:
Interaction energy shifts two-atom ground state ⇒ modified atomic tunnel rate.
2. Detection of molecules: measurement of tunneling atoms at a B field wheredeeply bound molecules do not tunnel.
A. Saenz: Confined few-body systems (41) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Measurement of the mean atom number
Positions for molecule formation: 776.01 G & 779.02 G
A. Saenz: Confined few-body systems (42) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Ab initio calculation
Exact diagonalization (full 6D) of Li2 Hamiltonian in a trap with experimentalparameters (varying scattering length with inner-wall shift).
Due to anisotropy (ωx 6= ωy ωz) two inelastic CIR (avoided crossings) expected.
A. Saenz: Confined few-body systems (43) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
More precise CIR detection (I)
Ramp B field non-adiabaticlly into region of avoided crossing:
coherent superposition of molecules and repulsive trap state (Rabi oscillation).
A. Saenz: Confined few-body systems (44) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
More precise CIR detection (II)
780.00 780.25 780.50 780.75 781.00
0.2
0.4
0.6
0.8
1.0
5 10 15 20 25 300.0
0.5
1.0
B0 = 780.49(0.01)G
Max
imum
frac
tion
of m
olec
ules
FWHM=0.25(0.03)G
Lorentzian fit
= 2 x 83 Hz
Frac
tion
of m
olec
ules
Hold time [ms]
B = 780.46G Damped Sine fit
Rabi frequency:
Ω = 1~
√W 2
n + δ2
Wn = 〈ψ(b) Φn|W |ψ0 Φ(0,0,0)〉
Variation of B:
allows fit of position
and width.
A. Saenz: Confined few-body systems (45) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Comparison ab initio result to experiment
COM Position [G] FWHM[G] Ω0[Hz]/ 2πexcitation exp. num. exp. num. exp. num.(2, 0, 0) 780.5 776.01 0.25(0.03) 0.35 83 64(0, 2, 0) 783.2 779.02 0.42(0.06)(∗) 0.35 75 (∗) 69
(∗) Magnetic field gradient B′ = 18.92 G/cm applied.
More details:
Sala, Zurn, Lompe, Wenz, Murmann, Serwane, Jochim, A.S.,
Phys. Rev. Lett. 110, 203202 (2013) .
A. Saenz: Confined few-body systems (46) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Universality of confinement-induced resonances:
Dipolar gases (heteronuclear molecules, Rydberg atoms):
Inelastic confinement-induced resonances seen in ab initio calculations.
They are tunable by varying the dipole-coupling strength!
[B. Schulz, S. Sala, and A. Saenz, New J. Phys. 17, 065002 (2015)]
A. Saenz: Confined few-body systems (47) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Universality of confinement-induced resonances:
Dipolar gases (heteronuclear molecules, Rydberg atoms):
Inelastic confinement-induced resonances seen in ab initio calculations.
They are tunable by varying the dipole-coupling strength!
[B. Schulz, S. Sala, and A. Saenz, New J. Phys. 17, 065002 (2015)]
Quantum-dot systems (electron pairs or excitons):
Inelastic confinement-induced resonances occur also for Coulomb interaction.
For electron pairs (no bound state) (smaller) change of density.
For excitons (electron-hole pairs) larger change of density.
A. Saenz: Confined few-body systems (47) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Universality of confinement-induced resonances:
Dipolar gases (heteronuclear molecules, Rydberg atoms):
Inelastic confinement-induced resonances seen in ab initio calculations.
They are tunable by varying the dipole-coupling strength!
[B. Schulz, S. Sala, and A. Saenz, New J. Phys. 17, 065002 (2015)]
Quantum-dot systems (electron pairs or excitons):
Inelastic confinement-induced resonances occur also for Coulomb interaction.
For electron pairs (no bound state) (smaller) change of density.
For excitons (electron-hole pairs) larger change of density.
−→ on-demand single-photon source!
[M. Troppenz, S. Sala, P.-I. Schneider, and A. Saenz submitted to Phys. Rev. B; arXiv:1509.01159]
A. Saenz: Confined few-body systems (47) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Summary
• Confinement effects and dimensionality are important.
A. Saenz: Confined few-body systems (48) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Summary
• Confinement effects and dimensionality are important.
• There are elastic and inelastic confinement-induced resonances.
A. Saenz: Confined few-body systems (48) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Summary
• Confinement effects and dimensionality are important.
• There are elastic and inelastic confinement-induced resonances.
• Inelastic CIRs allow for molecule formation, on-demand single-photon sources. . .
A. Saenz: Confined few-body systems (48) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Summary
• Confinement effects and dimensionality are important.
• There are elastic and inelastic confinement-induced resonances.
• Inelastic CIRs allow for molecule formation, on-demand single-photon sources. . .
• In 1D the two resonances overlap, in 2D the sign of asc differs.
A. Saenz: Confined few-body systems (48) 614. Heraeus Seminar, Bad Honnef, 20.04.2016
Summary
• Confinement effects and dimensionality are important.
• There are elastic and inelastic confinement-induced resonances.
• Inelastic CIRs allow for molecule formation, on-demand single-photon sources. . .
• In 1D the two resonances overlap, in 2D the sign of asc differs.
Inelastic CIRs and molecule formation:
see Phys. Rev. Lett. 109, 073201 (2012); Phys. Rev. Lett. 110, 203202 (2013)
Acknowledgments especially to:
Sergey Grishkevich, Simon Sala, Philipp-Immanuel Schneider (HU Berlin)
S. Jochim, G. Zurn, A. N. Wenz, S. Murmann, F. Serwane (Uni Heidelberg)
A. Saenz: Confined few-body systems (48) 614. Heraeus Seminar, Bad Honnef, 20.04.2016