Download The Remarkable Bose

The Remarkable Bose-Hubbard Dimer
David K. Campbell, Boston University
Winter School, August 2015
“Strongly Coupled Field Theories for Condensed Matter
and Quantum Information Theory”
International Institute of Physics, Natal
∧ = 0.5
∧=1.5
∧=5
Prolog: Axioms for Lecturers
•  Kac Axiom: “Tell the truth, nothing but the
truth, but NOT the whole truth.”
•  Weisskopf Axiom” “It is better to uncover
a little than to cover a lot.”
•  But we all tend to follow the Rubbia lemma:
•  TTN: What was the remark?
“Faster”
Bottom Lines
•  Temporal decoherence of BEC is critical for potential applications.
For a simple system (BEC dimer ~ Bosonic Josephson Junction=BJJ)
we show that coherence is enhanced near fixed points of a classical
analog system, including self-trapped fixed points but is supressed
near separatrix
•  We do this by introducing “global phase space” (GPS) portraits of
quantum observables including 1) the “condensate fraction”; and 2)
entanglement. We compare and contrast standard Gross-Pitaevski
equation (GPE) (mean field) with Bose-Hubbard model (fully quantum)
and show that much of the observed behavior can be understood by
studying the phase space of a related classical nonlinear dynamical
system, BUT
•  We predict novel quantum effects, possibly observable in near-term
experiments: 1) introducing dissipation (atom loss), we find surprising
result that coherence and entanglement can be enhanced by
dissipation; and 2) we predict quantum tunneling between selftrapped fixed points in the system
Outline
•  The Big Picture
–  Ultra-cold atoms (Bose-Einstein Condensates and Degenerate
Fermions)—key experimental breakthroughs
–  Some comments on BECs
•  Specific Background for talk
–  BECs in mean field description: continuum Gross-Pitaevski Eqn
–  BECs in Optical Lattices: discrete GPE ~ Discrete Nonlinear
Schrödinger Eqn.
–  BEC dimer= Bose Josephson Junction (BJJ)
•  Comparison of mean field results with experiments
•  The Bose Hubbard Model and quantum effects
–  “Global Phase space” study of quantum vs. mean field without and with
dissipation
–  Tunneling between self-trapped regions in the BH dimer
•  Summary and Conclusions
•  References
Ultracold atoms: Fermions vs. Bosons
810 nK
510 nK
240 nK
TTN: Which is which?
7Li Bosons
6Li Fermions
NB: Tc for BEC and TF for Fermions depend on densities, masses,
interactions. Typically scales as 1/M, for 87Rb atoms Tc ~ 300 nK.
Figure from Truscott, A., K. Strecker, W. McAlexander, G. Partridge, and R. G. Hulet, Science
291, 2570. (2001).
Bosons vs. Fermions
Figure from Duchon PhD thesis
Key Experimental Breakthroughs
•  Optical force on neutral atoms via AC Stark effect (induced
polarization): can change sign of force by using red-or bluedetuning of laser from dipole resonance.
•  Magneto-optical traps
•  “Optical molasses” means of slowing velocity (=cooling) atoms
(Nobel Prize 1997)
•  Evaporative cooling
•  Controlling size and sign of interactions via Feshbach resonances
•  Together methods can lower effective temperatures to
nanokelvin levels and offer exquisite control of interactions:
Good short summary in
B. deMarco and D.S. Jin, Science 285 1703-1706 (1999).
Review in I. Bloch et al., Rev Mod Phys 80, 885 (2008).
Background: BEC
•  Bose Einstein condensation
–  Macroscopic occupancy of single quantum state by large
number of identical bosons
–  Predicted in 1925 by Einstein for non-interacting bosons
–  Observed in 1995 by two groups, Wieman/Cornell and
Ketterle: Nobel Prize in 2001
Einstein’s Original Manuscript
Modeling BEC
BEC is described by a quantum wavefunction Φ(x,t). In the
context of cold atomic vapors trapped in an external potential
(typically combined magnetic confinement and optical potential)
in mean field approximation Φ(x,t) obeys the“Gross-Pitaevskii”
equation (GPE)
•  Nonlinear dynamicists recognize this immediately as a
variant of the Nonlinear Schrödinger equation, here in
3D and with an external potential—more on this later.
•  The interaction term—g0—describes s-wave scattering
of atoms
•  In 1D case (realizable depending on external potential)
we expect (continuum) solitons among excitations.
BECs in Optical Lattice
Counter-propagating laser pulses create standing wave
that interacts via AC Stark effect with neutral atoms, so
that the BEC experiences a periodic potential of the
form:
UL(x,y) is transverse confining potential, λ is the laser
wavelength (typically ~850 nm) and “z” is the direction
of motion.
Image from I. Bloch Nature 453 1016-1022 (2008)
BECs in Optical Lattices
•  Quantum system in periodic potential—just as in solid state. Can
describe in terms of (extended) Bloch wave functions or
(localized) Wannier wave functions, centered on the wells of the
potential (but allowing tunneling between the wells
•  Expanding the GPE in terms of Wannier functions leads to the
Discrete Nonlinear Schrodinger Equation (DNLSE) for the wave
function amplitudes. In normalization we shall later use
•  Comes from Hamiltonian (λ=2U/J)
BEC Dimer: Physical Realizations and Models
•  Single coherent BEC in a “double well” potential or two internal
(hyperfine) states with resonant interactions in single well.
–  Analogous to two-site “dimer” system and therefore called “BEC dimer” but
also to Josephson Junction therefore called Bosonic Josephson Junction
(BJJ).
–  Parameters: number of particles in BEC (N) and in each of two “wells”/states
(N1 and N2), z= N1-N2, phase difference between two condensates ( ϕ = ϕ1 –
ϕ2), resonant coupling/tunneling between two wells (J), interaction of Bosons
within condensate (U); single parameter in simple models Λ = (NU)/J
•  Two Models:
–  Gross-Pitaevskii equation (GPE): corresponds to integrable classical
dynamical system* in (z, ϕ) for this case
–  Bose-Hubbard Hamiltonian (BHH) : fully quantum *
* NB: Quantum dimer also integrable by Bethe Ansatz: solution not
useful for calculating physical observables
Different Physical realizations of Dimer
•  Two-well optical lattice
M. Albiez et al.,Phys. Rev. Lett. 95, 010402 (2005)
•  Two internal atomic states (87Rb)
T. Zibold et al. Phys. Rev. Lett. 105, 204101 (2010)
Mean-Field (GPE) Eqn for Dimer
Hamiltonian becomes
Operators replaced with c-numbers
Let Ψj = bj eiθj , where bj and θj are real. Define
z = [b12 − b22 ] / N = [N1 − N 2 ] / N, ϕ = θ 2 − θ1
The “Hamiltonian” becomes
Equivalent Nonlinear Dynamical System
TTN: What kind of dynamical system is this?
Answer: a simple one degree of freedom (=>integrable)
dynamical like a pendulum but with length dependent on its
momentum.
Hamilton’s equations become
z = − 1− z sin ϕ
z cosϕ
ϕ = Λz +
1− z 2
2
Equivalent Nonlinear Dynamical System
Fixed Points ? Work out on blackboard
Further analysis of FPs shows for
Λ < 1, two stable FPs, (z,φ)= (0,0) and (0,π)
then a “bifurcation at Λ = 1
and for
Λ > 1, three stable FPs, (z,φ)=(0,0), (z+,π), and (z-,π)
z+/-= +/- (Λ2 – 1)1/2/Λ ≈ +/- 1 for Λ => ∞
Images of ”Global Phase Space”
Λ = 0.5
Λ= 1.5
Λ=5
Comparison to Experiments !
Next slides show how dynamics of the dimer/BJJ
observed in experiments follows (or does not follow)
classical phase portraits of this system.
Experimental Observations
Bifurcation at Λ = 1
T. Zibold et al. Phys. Rev. Lett. 105, 204101 (2010)
Experimental Observations
Existence of Zc for self-trapping: work out on board
Z below Zc(Λ)
M. Albiez et al, Phys. Rev. Lett. 95 010402 (2005)
Z above Zc(Λ)
Experimental Observations
Here Zc ~ ½ =>
Λ~15
M. Albiez et al, Phys. Rev. Lett. 95 010402 (2005)
Experimental Observations
Full Global Phase Space plots
NB: Each point is different experiment!
T. Zibold et al., PRL 105 204101 (2010)
Experimental Observations
View on Bloch sphere
T. Zibold et al. Phys. Rev. Lett. 105, 204101 (2010)
Full Quantum Dynamics Bose-Hubbard Model
hopping
interaction (repulsive in our case)
•  Atoms in a double-well optical trap
•  Two spin states of atoms trapped in one well
•  Observables of interest:
Single-particle density matrix (Condensate Fraction/ purity)
Entanglement measure
Observables and Results
•  We study the time evolution of quantum observables: discuss two today
1) “condensate fraction”—C(z,ϕ) ≅ λmax /N ≤ 1, λmax = largest
eigenvalue of single particle density matrix
2) “entanglement”— E ≅ |<a1*a2>|2 -<a1*a1a2*a2>, E > 0 for entangled
state
•  Results
–  Both C(z,ϕ) and E generally “track” classical orbits in GPS
–  C(z,ϕ) decreases dramatically near classical separatrix, remains large at
classical fixed point and especially in self-trapped region=> regular
motion enhances coherence, chaotic motion destroys coherence
–  Symmetry z => - z of classical equations is broken by quantum dynamics
–  “Ridge” of enhanced coherence exists in quantum but not in LD or GPE
(classical) or dynamics
–  Overlap of initial coherent state with eigenfunctions of BHH is minimal
along separatrix, high near fixed points
Classical phase space, quantum dynamics
H. Hennig et al., PRA 86, 051604(R) (2012)
Condensate Fraction at T=1 sec Λ = 1.5
Note considerable variation in C(z,ϕ,T=1) and how it generally
follows contours of classical dynamics
Constant energy contours
Separatrix
Orbits around FP
(0,0): have <z>=<ϕ>=0
z=>-z symmetry broken
Orbits around FP+ and FP- are macroscopic self-trapped states, <z>≠0, <ϕ>=π
“π-phase” orbits still exist, <z>=0, <ϕ>=π
Results for C of Dimer Λ = 5.0
GPE=“classical” gives C=1
ϕ = 0.85
Separatrix,
C falls rapidly
nearby: decoherence
enhanced
Separatrix,
C falls, then
recovers as
FP(0,0)
“controls”
dynamics
“Ridge” of increased
C, pure quantum
effect
ϕ = 0.0
Initial State overlap with Eigenfunctions
Color code shows
A(ϕ,z)≅maxn<ϕ,z|En>, where
En is an eigenstate of the BH
dimer
Λ = 5.0: Λ = 1.5
Conclusion: localization
in phase space (FPs)
maintains coherence,
delocalization
(separatrix) induces
decoherence.
What do quantum orbits really look like?
Large Λ, near fixed point, green is classical orbit
Dissipationless Quantum Dynamics
TTN: What do you see?
ANS: Two frequencies
Quantum „orbits,” interesting features
„thick” orbits
two frequencies in EPR,
condensate fraction
Explain two frequencies: High frequency
High frequency is mean-field (semiclassical)
Conversion to seconds
• Data from power spectrum of EPR:
Low-frequency oscillations
Low frequency quantum revival
In the limit J/U = 0, for any state
and
:
But revivals seen also for J/U > 0 (or, Λ < ∞)
Greiner et al., Nature 419 51
(2002)
Summary: Dynamics near the FP
High frequency is mean-field (semiclassical)
Low frequency is a quantum revival:
But how far from the fixed
point is this a good picture?
Low-Λ anomaly: Husimi density
TTN: What do you notice for Λ~ 1 ?
ANS: Looks like quantum
tunneling between STFPs!!
Projections as Interpretation of Two Frequencies
Energy eigenstate expansion:
Projection onto „most important” states:
Three eigenstates produce beats in observables:
Replacing the full state with the projection should work well where,
How informative is the projection?
2 states: only fast motion
3 states: fast and slow motion
How informative is the projection?
How informative is the projection?
How informative is the projection?
For z = 0.95, Φ = π the contributions of three
highest eigenstates are
a0 = 0.9353,
a1 = 0.3474,
a2 = 0.0653 so that
|a0|2 + |a1|2 + |a2|2 = 0.9997
And eigenfrequency differences fit with
observed frequencies in numerics almost to
within 3 %
Dissipation in a BEC dimer
•  Focused electron beam removes atoms from one
of the wells
Gericke et al., Nature Phys. 4, 949 (2008)
•  Open quantum system: quantum jump method
Dalibard et al., PRL 68(5) 580 (1992)
Garraway and Knight, PRA 49(2) 1266 (1994)
Witthaut et al., PRA 83(6) 1 (2011)
Dissipation-induced coherence
Dissipative Quantum Dynamics
Λ=5
Results: phase space
TTN: What did dissipation do?
Ans: Drove system near fixed point!
Results: condensate fraction
Results: EPR entanglement
What about tunneling for Λ ~ 1 ?
Classical approximation poor for Λ close to 1; can we observe quantum
tunneling in experiment ???
Summary of Main Results Thus far
Results for 1) “condensate fraction”—C(z,ϕ) ≅ λmax /N ≤ 1, λmax = largest
eigenvalue of single particle density matrix and “entanglement”—
E ≅ |<a1*a2>|2 -<a1*a1a2*a2>, E > 0 for entangled state
–  Both C(z,ϕ) and E generally “track” classical orbits in GPS
–  C(z,ϕ) decreases dramatically near classical separatrix, remains large at
– 
– 
– 
– 
classical fixed point and especially in self-trapped region=> regular
motion enhances coherence, chaotic motion destroys coherence
Overlap of initial coherent state with eigenfunctions of BHH is minimal
along separatrix, high near fixed points
Quantum “orbits” show two frequencies: one semiclassical, the other
quantum revival ; interpretation in terms of three leading
eigenfrequencies
Dissipation can enhance coherence for both C and E
Predict tunneling between self-trapped FPs for Λ ~ 1 + ε
Husimi function vs Time
Behavior of Husimi function vs. time: tunneling between selftrapped fixed points is clearly seen: Λ = 1.1, J= 10 Hz
Interpretation ?
Near the self-trapped fixed points (STFP) we have seen
that only two states could explain most of dynamics:
does this work here?
YES! Energy difference between highest two states in
BH model corresponds very closely to tunneling
frequency: think of symmetric and anti-symmetric
states in simple 1D QM.
Can work this out in more detail with semi-classical
approach to tunneling: full details in T. Pudlik et al.
Phys.Rev A 90, 053610 (2014).
Probability of states vs. NB
Tunneling frequency vs. NB
Observation in experiment?
Prediction of tunneling frequency vs Λ for experimental parameters
of T. Zibold et al. Phys Rev Lett 105 204101 (2010)
(U= 2π (x) 0.063Hz, N=500
The Dimer and beyond
For dimer, key challenge is experimental observation
of tunneling between self-trapped fixed points.
For spatially extended systems, “Intrinsic Localized
Modes”/ “Discrete Breathers” play role of self-trapped
states: may serve as means of maintain quantum
coherence in large optical lattices or macromolecules.
ILMs/DBs in Spatially extended systems
Results of Ng et al (Geisel group) confirm
for extended systems (M ~ 300) that DBs
are not formed at small Λ (=λN) but are
readily formed at large Λ.
Calculations are within the mean field/GPE
model
Fig 2 from Ng et al., New J. Phys 11, 073045 (2009).
Bottom Lines Redux
•  In BEC dimer=BJJ coherence effects (condensate fraction.,
entanglement) are enhanced near fixed points of a classical analog
system, including self-trapped fixed points but are suppressed near
separatrix.
•  Introducing dissipation (atom loss), we find surprising result that
coherence and entanglement can be enhanced !
•  We predict quantum tunneling between self-trapped fixed points in
the system.
•  We speculate that similar effects may occur in extended optical
lattices due for formation of Intrinsic Localized Modes/Discrete
Breathers.
Questions ?
References
•  Excellent review article on theory and experiments, I.
Bloch et al. Rev. Mod. Phys. 80, 885-964 (2008).
•  H. Hennig, D. Witthaut, and DKC, “Global phase space of
coherence and entanglement in a double-well BoseEinstein Condensate, Phys. Rev. A 86 , 051604(R)
(2012).
•  T. Pudlik, H. Hennig, D. Witthaut, and DKC “Dynamics of
entanglement in a dissipative Bose-Hubbard Dimer,”
Phys. Rev. A 88, 063606 (2013).
•  T. Pudlik, H. Hennig, D. Witthaut, and DKC “Tunneling in
the self-trapped regime of a two-well Bose-Einstein
condensate” Phys. Rev. A 90, 053610 (2014)
END