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Coherent backscattering in the Fock space of a disordered - - PowerPoint PPT Presentation

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Coherent backscattering in the Fock space of a disordered Bose-Hubbard system Peter Schlagheck 20/3/2015 Coherent backscattering in Fock space Phys. Rev. Lett. 112, 140403 (2014) Coworkers Thomas Engl Juan Diego Urbina Klaus Richter

slide-1
SLIDE 1

Coherent backscattering in the Fock space of a disordered Bose-Hubbard system

Peter Schlagheck 20/3/2015

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-2
SLIDE 2

Coworkers

Thomas Engl (Regensburg) Juan Diego Urbina (Regensburg) Klaus Richter (Regensburg) Julien Dujardin (Li` ege) Arturo Arg¨ uelles (now in Cali)

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-3
SLIDE 3

Outline

Introduction to coherent backscattering (CBS) Semiclassical theory of Bose-Hubbard systems Numerical results for the backscattering probability Implication for quantum thermalization Proposal for an experimental verification Conclusion

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-4
SLIDE 4

Outline

Proposal for an experimental verification Numerical results for the backscattering probability Implication for quantum thermalization Introduction to coherent backscattering (CBS) Semiclassical theory of Bose-Hubbard systems Conclusion

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-5
SLIDE 5

Proposal for a many-body CBS experiment

Consider an isolated 2D sheet of a 3D optical lattice . . .

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-6
SLIDE 6

Proposal for a many-body CBS experiment

Consider an isolated 2D sheet of a 3D optical lattice within which you isolate a single plaquette (e.g. by means of a focused red-detuned laser beam)

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-7
SLIDE 7

Proposal for a many-body CBS experiment

Consider an isolated 2D sheet of a 3D optical lattice within which you isolate a single plaquette (e.g. by means of a focused red-detuned laser beam)

  • L. Tarruell et al., Nature 483, 302 (2012)

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-8
SLIDE 8

Proposal for a many-body CBS experiment

Experimental procedure:

  • 1. Load the lattice with a well-defined number of (bosonic)

atoms in the deep Mott-insulator regime

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-9
SLIDE 9

Proposal for a many-body CBS experiment

  • 2. Add some disorder (by means of an optical speckle field)

and/or randomly displace the focus of the red-detuned laser beam

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-10
SLIDE 10

Proposal for a many-body CBS experiment

  • 2. Add some disorder (by means of an optical speckle field)

and/or randomly displace the focus of the red-detuned laser beam

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-11
SLIDE 11

Proposal for a many-body CBS experiment

  • 3. Switch on the inter-site hopping and let the atoms move . . .

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-12
SLIDE 12

Proposal for a many-body CBS experiment

  • 3. Switch on the inter-site hopping and let the atoms move . . .

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-13
SLIDE 13

Proposal for a many-body CBS experiment

  • 4. Quench back to the Mott regime after a given evolution

time and detect the atomic population on each site

  • W. Bakr et al., Nature 462, 74 (2009)
  • J. Sherson et al., Nature 467, 68 (2010)
  • S. F¨
  • lling et al., Nature 448, 1029 (2007) (Brillouin zone mapping)

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-14
SLIDE 14

Proposal for a many-body CBS experiment

3 0 0 3 0 1

  • 4. Quench back to the Mott regime after a given evolution

time and detect the atomic population on each site

  • W. Bakr et al., Nature 462, 74 (2009)
  • J. Sherson et al., Nature 467, 68 (2010)
  • S. F¨
  • lling et al., Nature 448, 1029 (2007) (Brillouin zone mapping)

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-15
SLIDE 15

Proposal for a many-body CBS experiment

3 0 0 3 0 1

  • 5. Repeat the experiment with the same initial state but for a

different disorder configuration

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-16
SLIDE 16

Proposal for a many-body CBS experiment

3 0 0 3 0 1

  • 5. Repeat the experiment with the same initial state but for a

different disorder configuration

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-17
SLIDE 17

Proposal for a many-body CBS experiment

3 0 0 3 0 1 1 0 3 2 0 1

  • 5. Repeat the experiment with the same initial state but for a

different disorder configuration

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-18
SLIDE 18

Proposal for a many-body CBS experiment

3 0 0 3 0 1 1 0 3 2 0 1

  • 5. Repeat the experiment with the same initial state but for a

different disorder configuration

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-19
SLIDE 19

Proposal for a many-body CBS experiment

3 0 0 3 0 1 1 0 3 2 0 1

  • 5. Repeat the experiment with the same initial state but for a

different disorder configuration

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-20
SLIDE 20

Proposal for a many-body CBS experiment

3 0 0 3 0 1 1 0 3 2 0 1 2 1 4 0 0 0

  • 5. Repeat the experiment with the same initial state but for a

different disorder configuration

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-21
SLIDE 21

Proposal for a many-body CBS experiment

3 0 0 3 0 1 1 0 3 2 0 1 2 1 4 0 0 0 2 1 2 1 1 0

  • 5. Repeat the experiment with the same initial state but for a

different disorder configuration

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-22
SLIDE 22

Proposal for a many-body CBS experiment

3 0 0 3 0 1 1 0 3 2 0 1 2 1 4 0 0 0 2 1 2 1 1 0 3 1 0 1 1 1

  • 5. Repeat the experiment with the same initial state but for a

different disorder configuration

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-23
SLIDE 23

Proposal for a many-body CBS experiment

3 0 0 3 0 1 1 0 3 2 0 1 2 1 4 0 0 0 2 1 2 1 1 0 3 1 0 1 1 1 0 1 2 1 2 1

  • 5. Repeat the experiment with the same initial state but for a

different disorder configuration

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-24
SLIDE 24

Proposal for a many-body CBS experiment

3 0 0 3 0 1 1 0 3 2 0 1 2 1 4 0 0 0 2 1 2 1 1 0 3 1 0 1 1 1 0 1 2 1 2 1 Average population per site: 7/6 = 1.167 . . . but we are now interested in the full statistical information

  • f the experimental outcomes

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-25
SLIDE 25

Proposal for a many-body CBS experiment

General expectation from quantum statistical physics: − → all quantum states that have about the same total energy − → as the initial state are equally likely to be detected

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-26
SLIDE 26

Proposal for a many-body CBS experiment

General expectation from quantum statistical physics: − → all quantum states that have about the same total energy − → as the initial state are equally likely to be detected

011122 012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110

final state

5 10

Statistics after 1000 measurements

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-27
SLIDE 27

Proposal for a many-body CBS experiment

General expectation from quantum statistical physics: − → all quantum states that have about the same total energy − → as the initial state are equally likely to be detected

011122 012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110

final state

5 10

Statistics after 2000 measurements

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-28
SLIDE 28

Proposal for a many-body CBS experiment

General expectation from quantum statistical physics: − → all quantum states that have about the same total energy − → as the initial state are equally likely to be detected

011122 012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110

final state

10 20

Statistics after 5000 measurements

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-29
SLIDE 29

Proposal for a many-body CBS experiment

General expectation from quantum statistical physics: − → all quantum states that have about the same total energy − → as the initial state are equally likely to be detected

011122 012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110

final state

50

Statistics after 10000 measurements

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-30
SLIDE 30

Proposal for a many-body CBS experiment

General expectation from quantum statistical physics: − → all quantum states that have about the same total energy − → as the initial state are equally likely to be detected

011122 012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110

final state

50 100

Statistics after 20000 measurements

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-31
SLIDE 31

Proposal for a many-body CBS experiment

General expectation from quantum statistical physics: − → all quantum states that have about the same total energy − → as the initial state are equally likely to be detected

011122 012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110

final state

100 200

Statistics after 50000 measurements

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-32
SLIDE 32

Proposal for a many-body CBS experiment

General expectation from quantum statistical physics: − → all quantum states that have about the same total energy − → as the initial state are equally likely to be detected

011122 012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110

final state

500

Statistics after 100000 measurements

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-33
SLIDE 33

Proposal for a many-body CBS experiment

General expectation from quantum statistical physics: − → all quantum states that have about the same total energy − → as the initial state are equally likely to be detected

011122 012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110 0.005

Detection probability This is not the case for the initial state which is twice as often detected as other states with comparable total energy

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-34
SLIDE 34

Proposal for a many-body CBS experiment

General expectation from quantum statistical physics: − → all quantum states that have about the same total energy − → as the initial state are equally likely to be detected

011122 012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110 0.005

Detection probability This is not the case for the initial state which is twice as often detected as other states with comparable total energy − → signature of coherent backscattering in Fock space

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-35
SLIDE 35

Coherent backscattering in disordered systems

→ constructive wave interference between reflected classical paths and their time-reversed counterparts

  • θ
  • 0.4π
  • 0.2π

0.2π 0.4π 1 2

backscattered current angle θ Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-36
SLIDE 36

Coherent backscattering in disordered systems

→ constructive wave interference between reflected classical paths and their time-reversed counterparts

  • θ

Coherent backscattering of laser light in disordered media

  • M. P

. Van Albada and A. Lagendijk, PRL 55, 2692 (1985) P .-E. Wolf and G. Maret, PRL 55, 2696 (1985)

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-37
SLIDE 37

Coherent backscattering in disordered systems

→ constructive wave interference between reflected classical paths and their time-reversed counterparts

  • θ

Coherent backscattering of laser light in disordered media

  • M. P

. Van Albada and A. Lagendijk, PRL 55, 2692 (1985) P .-E. Wolf and G. Maret, PRL 55, 2696 (1985)

Coherent backscattering of ultracold atoms in 2D disorder

  • F. Jendrzejewski et al., PRL 109, 195302 (2012)

(see also N. Cherroret et al., PRA 85, 011604(R) (2012) )

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-38
SLIDE 38

Coherent backscattering in disordered systems

→ constructive wave interference between reflected classical paths and their time-reversed counterparts

  • θ

Coherent backscattering of laser light in disordered media

  • M. P

. Van Albada and A. Lagendijk, PRL 55, 2692 (1985) P .-E. Wolf and G. Maret, PRL 55, 2696 (1985)

Coherent backscattering of ultracold atoms in 2D disorder

  • F. Jendrzejewski et al., PRL 109, 195302 (2012)

(see also N. Cherroret et al., PRA 85, 011604(R) (2012) ) → generalization to interacting many-body systems?

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-39
SLIDE 39

Many-body CBS in disordered Bose-Hubbard rings

ˆ H =

L

  • l=1
  • Elˆ

a†

l ˆ

al − J

  • ˆ

a†

l ˆ

al−1 + ˆ a†

l−1ˆ

al

  • + U

2 ˆ a†

l ˆ

a†

l ˆ

alˆ al

  • J

J J J J J E1 E2 E3 E4 E5 E6

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-40
SLIDE 40

Many-body CBS in disordered Bose-Hubbard rings

ˆ H =

L

  • l=1
  • Elˆ

a†

l ˆ

al − J

  • ˆ

a†

l ˆ

al−1 + ˆ a†

l−1ˆ

al

  • + U

2 ˆ a†

l ˆ

a†

l ˆ

alˆ al

  • J

J J J J J E1 E2 E3 E4 E5 E6

Classical description: discrete Gross-Pitaevskii equation i ∂ ∂tψl(t) = Elψl(t)−J [ψl+1(t) + ψl−1(t)]+U

  • |ψl(t)|2 − 1
  • ψl(t)

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-41
SLIDE 41

Many-body CBS in disordered Bose-Hubbard rings

Semiclassical van Vleck-Gutzwiller theory: − → Represent the quantum transition amplitude nf| ˆ U|ni ≡ nf| exp[− i

t ˆ

H]|ni =

γ AγeiRγ/

− → in terms of classical (Gross-Pitaevskii) trajectories γ going − → from ψl(0) =

  • ni

l + 0.5 eiθi

l to ψl(t) =

  • nf

l + 0.5 eiθf

l

− → for all l = 1, . . . , L with some arbitrary phases 0 ≤ θi/f

l

< 2π |ni(f) ≡ |ni(f)

1

. . . ni(f)

L : initial (final) Fock state on the ring

Rγ = classical action of the trajectory γ Aγ = stability amplitude (related to Lyapunov exponent) of γ

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-42
SLIDE 42

Many-body CBS in disordered Bose-Hubbard rings

Semiclassical van Vleck-Gutzwiller theory: − → Represent the quantum transition amplitude nf| ˆ U|ni ≡ nf| exp[− i

t ˆ

H]|ni =

γ AγeiRγ/

− → in terms of classical (Gross-Pitaevskii) trajectories γ going − → from ψl(0) =

  • ni

l + 0.5 eiθi

l to ψl(t) =

  • nf

l + 0.5 eiθf

l

− → for all l = 1, . . . , L with some arbitrary phases 0 ≤ θi/f

l

< 2π Average detection probability of the Fock state |nf: |nf| ˆ U|ni|2 =

γ,γ′ AγAγ′ei(Rγ−Rγ′)/

  • nf| ˆ

U|ni|2 =

γ,γ′ AγAγ′ = 0 if Rγ = Rγ′

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-43
SLIDE 43

Many-body CBS in disordered Bose-Hubbard rings

Semiclassical van Vleck-Gutzwiller theory: − → Represent the quantum transition amplitude nf| ˆ U|ni ≡ nf| exp[− i

t ˆ

H]|ni =

γ AγeiRγ/

− → in terms of classical (Gross-Pitaevskii) trajectories γ going − → from ψl(0) =

  • ni

l + 0.5 eiθi

l to ψl(t) =

  • nf

l + 0.5 eiθf

l

− → for all l = 1, . . . , L with some arbitrary phases 0 ≤ θi/f

l

< 2π Average detection probability of the Fock state |nf: |nf| ˆ U|ni|2 =

γ |Aγ|2 if nf = ni

|ni| ˆ U|ni|2 = 2

γ |Aγ|2 due to CBS

in the presence of chaos (ergodicity)

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-44
SLIDE 44

Many-body CBS in disordered Bose-Hubbard rings

Semiclassical van Vleck-Gutzwiller theory: − → Represent the quantum transition amplitude nf| ˆ U|ni ≡ nf| exp[− i

t ˆ

H]|ni =

γ AγeiRγ/

− → in terms of classical (Gross-Pitaevskii) trajectories γ going − → from ψl(0) =

  • ni

l + 0.5 eiθi

l to ψl(t) =

  • nf

l + 0.5 eiθf

l

− → for all l = 1, . . . , L with some arbitrary phases 0 ≤ θi/f

l

< 2π Average detection probability of the Fock state |nf: |nf| ˆ U|ni|2 =

γ |Aγ|2 if nf = ni

|ni| ˆ U|ni|2 = 2

γ |Aγ|2 due to CBS

ni n

f

= nf n

i

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-45
SLIDE 45

Comparison with numerical data

ˆ H =

L

  • l=1
  • Elˆ

a†

l ˆ

al − J

  • ˆ

a†

l ˆ

al−1 + ˆ a†

l−1ˆ

al

  • + U

2 ˆ a†

l ˆ

a†

l ˆ

alˆ al

  • 011122

012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110 0.005

quantum

Detection probability

U = J 0 ≤ El ≤ 2J

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-46
SLIDE 46

Comparison with numerical data

ˆ H =

L

  • l=1
  • Elˆ

a†

l ˆ

al − J

  • ˆ

a†

l ˆ

al−1 + ˆ a†

l−1ˆ

al

  • + U

2 ˆ a†

l ˆ

a†

l ˆ

alˆ al

  • 011122

012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110 0.005

quantum classical

Detection probability

U = J 0 ≤ El ≤ 2J

nf| ˆ U|ni|2classical =

  • γ

|Aγ|2 = 2π dθi

2

2π · · · 2π dθi

L

2π

L

  • l=2

δ

  • nf

l + 0.5 − |ψl(t; ni 1, 0, ni 2, θi 2 . . . ni L, θi L)|2

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-47
SLIDE 47

Comparison with numerical data

ˆ H =

L

  • l=1
  • Elˆ

a†

l ˆ

al − J

  • ˆ

a†

l ˆ

al−1 + ˆ a†

l−1ˆ

al

  • + U

2 ˆ a†

l ˆ

a†

l ˆ

alˆ al

  • 223334

232334 233243 233432 242333 322334 323243 323432 332234 332342 333242 334332 343223 423233 432323 433322 0.005

quantum classical

Detection probability

U = 4J 0 ≤ El ≤ 10J

nf| ˆ U|ni|2classical =

  • γ

|Aγ|2 = 2π dθi

2

2π · · · 2π dθi

L

2π

L

  • l=2

δ

  • nf

l + 0.5 − |ψl(t; ni 1, 0, ni 2, θi 2 . . . ni L, θi L)|2

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-48
SLIDE 48

Comparison with numerical data

ˆ H =

L

  • l=1
  • Elˆ

a†

l ˆ

al − J

  • ˆ

a†

l ˆ

al−1 + ˆ a†

l−1ˆ

al

  • + U

2 ˆ a†

l ˆ

a†

l ˆ

alˆ al

  • 011122

012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110 0.005

quantum

Detection probability

U = J 0 ≤ El ≤ 2J

Ultimate experimental verification of CBS: − → break time-reversal invariance by a synthetic gauge field − → Y.-J. Lin et al., Nature 462, 628 (2009) − → J. Struck et al., Science 333, 996 (2011)

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-49
SLIDE 49

Comparison with numerical data

ˆ H =

L

  • l=1
  • Elˆ

a†

l ˆ

al − J

  • ˆ

a†

l ˆ

al−1eiφ + ˆ a†

l−1ˆ

ale−iφ + U 2 ˆ a†

l ˆ

a†

l ˆ

alˆ al

  • eiφ

011122 012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110 0.005

quantum

Detection probability

U = J 0 ≤ El ≤ 2J

Ultimate experimental verification of CBS: − → break time-reversal invariance by a synthetic gauge field − → Y.-J. Lin et al., Nature 462, 628 (2009) − → J. Struck et al., Science 333, 996 (2011)

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-50
SLIDE 50

Comparison with numerical data

ˆ H =

L

  • l=1
  • Elˆ

a†

l ˆ

al − J

  • ˆ

a†

l ˆ

al−1eiφ + ˆ a†

l−1ˆ

ale−iφ + U 2 ˆ a†

l ˆ

a†

l ˆ

alˆ al

  • eiφ

011122 012121 021211 101221 110122 111202 112102 112210 121012 121201 122110 202111 211012 211201 212110 221110 0.005

quantum quantum, φ = π/8

Detection probability

U = J 0 ≤ El ≤ 2J

Ultimate experimental verification of CBS: − → break time-reversal invariance by a synthetic gauge field − → Y.-J. Lin et al., Nature 462, 628 (2009) − → J. Struck et al., Science 333, 996 (2011)

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)
slide-51
SLIDE 51

Some conclusions

Coherent backscattering in the Fock space of a Bose-Hubbard system privileges, on average, the initial Fock state as compared to other states with comparable energy, significantly affects quantum ergodicity in finite systems, even if the classical dynamics is fully ergodic, can be experimentally detected with ultracold atoms, relies on time-reversal invariance and can therefore be switched off with a synthetic gauge field,

  • T. Engl, J. Dujardin, A. Arg¨

uelles, P .S., K. Richter, and J. D. Urbina,

  • Phys. Rev. Lett. 112, 140403 (2014)

Coherent backscattering in Fock space

  • Phys. Rev. Lett. 112, 140403 (2014)