quantum probing of bose-hubbard model via memory
TRANSCRIPT
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QUANTUM PROBING OF BOSE-HUBBARD MODEL VIA MEMORY EFFECTS
Francesco Cosco Turku Centre for Quantum Physics, University of Turku
Non Markovian Quantum Dynamics Cortona, 26 August 2015
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Ø Models: Environment and Probe
Ø Bose-Hubbard Model: Ø SF phase, Bogoliubov theory Ø MI phase, Three states theory
Ø Probe: Impurity Atom
Ø Results Ø Decoherence Fucntion Ø Non-Markovianity (BLP measure) Ø Density fluctuations and Density-density
correlation functions
Ø Contents
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Ø Expansion in Wannier States Ø Bose-Hubbard Model
Ø Parameters
Ø Bosonic System
�(x) =X
i
X
n
!
ni (x)c
ni
HBH = �JX
i
(c†i+ici + c†i ci+1) +U
2
X
i
ni(ni � 1)� µX
i
ni
U = g
Zdx!
0⇤i (x)!0⇤
i (x)!0i (x)!
0i (x)
H =
Zdx �(x)†h(x)�(x) +
g
2
Zdx �(x)†�(x)†�(x)�(x)
J = �Z
dx!
0⇤i (x)h(x)!0
i+1(x)
’
’
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On site interaction
U << J U >> J U ! 1Superfluid Mott Insulator Free Fermions
[1] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and Daniel S. Fisher, Phys. Rev. B 40, 546 (1989) [2] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, Rev. Mod. Phys. 83, 1405 (2011)
| GS(J = 0)i =NsY
i
(c†i )g
pg!
|vaci
| GS(J = 0)i =(a†k=0)
N
pN !
|vaci
HBH = �JX
i
(c†i+ici + c†i ci+1) +U
2
X
i
ni(ni � 1)� µX
i
ni
U
Ø Ground-states
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Probe: Impurity Atom
(x) =1p2⇡s
e
� x
2
2s2
Hint = g |ei he|⌦Z
dx | (x)|2n(x)
Ug = g
Zdx| (x)!0(x)|2
Hint = Ug |ei he| c†0c0
Hint = g |ei he|⌦Z
dx | (x)|2|!0(x)|2c†0c0
n0 = n0 + ˆ�n0
Ø Localized Probe
Ø Density-density interaction
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Ø J>>U Superfluid Phase Ø Bogoliubov Theory3
{|ni , |n± 1i}i=1...Ns
[3] D. van Oosten, P. van der Straten, and H. T. C. Stoof, Phys. Rev. A 63, 053601 (2001) [4] P. Barmettler, D. Poletti, M. Cheneau, and C. Kollath, Phys. Rev. A 85, 053625 (2012)
HBH =X
k
!k b†k bk
H =X
k,�=±!k,��
†k,��k,�
Ø SF vs MI
n
n+n�
ˆ�n0 =X
k
�k
Ns(b†k + bk)
ˆ�n0 = n0,+ � n0,�
Ø J<<U Mott Insulator Phase Ø Three States Theory4
ak=0 !pN0 + �ak=0
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Ø Probe dynamics
[5] Heinz-Peter Breuer, Elsi-Mari Laine, and Jyrki Piilo, Phys. Rev. Lett. 103, 210401 [6] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, arXiv:1505.01385 (2015)
N = max
⇢1,2(0)
Z
�>0�(t, ⇢1,2(0))
�(t) = �Z t
0dt1 �(t1)
�(t) = U2g Re
Z t
0dt1 Tr[ ˆ�n0(t1) ˆ�n0(0)⇢E ]
⇢s(t) =
✓⇢gg(0) ⇢eg(0)e�(t)+i�(t)
⇢ge(0)e�(t)�i�(t) ⇢ee(0)
◆
Ø Dephasing rate and decoherence function
Ø Non-Markovianity measure5
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N
�(t)|⇢eg(t)|
Ø Results SF
Time
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�(t)
h ˆ�ni(t)i
Ø Results SF
Time
i�site
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h ˆ�ni(t) ˆ�n0(0)i
Ø Results SF
Time
i�site
i�site
Re
Im
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Ø Results MI
N�(t)
|⇢eg(t)|
TimeTime
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Conclusions
Ø The decoherence in the open system dynamics detects features of the many-body environment
Ø Candidate as non-destructive quantum probe for ultracold atoms
Ø Ongoing work Ø Correlations and density fluctuations in the Mott
Phase
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THANK YOU FOR YOUR ATTENTION!
Non Markovian Quantum Dynamics Cortona, 26 August 2015