The Remarkable Bose-Hubbard Dimer

∧ = 0.5 ∧=1.5 ∧=5

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

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 self-trapped 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

7Li Bosons 6Li Fermions

810 nK

510 nK

240 nK

TTN: Which is which?

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 blue-detuning 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

Image from I. Bloch Nature 453 1016-1022 (2008)

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.

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

T. Zibold et al. Phys. Rev. Lett. 105, 204101 (2010)

•  Two-well optical lattice

•  Two internal atomic states (87Rb) M. Albiez et al.,Phys. Rev. Lett. 95, 010402 (2005)

Mean-Field (GPE) Eqn for Dimer

Operators replaced with c-numbers

Hamiltonian becomes

z = [b12 − b2

2 ] / N = [N1 − N2 ] / N , ϕ = θ2 −θ1

Let Ψj = bj eiθj , where bj and θj are real. Define

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− z2 sinϕ

ϕ = Λz + zcosϕ1− z2

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

T. Zibold et al. Phys. Rev. Lett. 105, 204101 (2010)

Bifurcation at Λ = 1

Experimental Observations

M. Albiez et al, Phys. Rev. Lett. 95 010402 (2005)

Z below Zc(Λ) Z above Zc(Λ)

Existence of Zc for self-trapping: work out on board

Experimental Observations

M. Albiez et al, Phys. Rev. Lett. 95 010402 (2005)

Here Zc ~ ½ => Λ~15

Experimental Observations

T. Zibold et al., PRL 105 204101 (2010)

Full Global Phase Space plots

NB: Each point is different experiment!

Experimental Observations

T. Zibold et al. Phys. Rev. Lett. 105, 204101 (2010)

View on Bloch sphere

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

Orbits around FP (0,0): have <z>=<ϕ>=0

Separatrix

Orbits around FP+ and FP- are macroscopic self-trapped states, <z>≠0, <ϕ>=π

z=>-z symmetry broken

“π-phase” orbits still exist, <z>=0, <ϕ>=π

Constant energy contours

Results for C of Dimer Λ = 5.0

ϕ = 0.85

ϕ = 0.0

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

GPE=“classical” gives C=1

Initial State overlap with Eigenfunctions

Color code shows A(ϕ,z)≅maxn<ϕ,z|En>, where En is an eigenstate of the BH dimer Conclusion: localization

in phase space (FPs) maintains coherence, delocalization (separatrix) induces decoherence.

Λ = 5.0: Λ = 1.5

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 :

Greiner et al., Nature 419 51 (2002)

But revivals seen also for J/U > 0 (or, Λ < ∞)

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

Three eigenstates produce beats in observables:

Energy eigenstate expansion: Projection onto „most important” states:

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

•  Open quantum system: quantum jump method

Gericke et al., Nature Phys. 4, 949 (2008)

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 self-trapped 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 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.

For dimer, key challenge is experimental observation of tunneling between self-trapped fixed points.

ILMs/DBs in Spatially extended systems

Fig 2 from Ng et al., New J. Phys 11, 073045 (2009).

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

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 Bose-Einstein 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