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

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

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)

’

’

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

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

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

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

N

�(t)|⇢eg(t)|

Ø  Results SF

Time

�(t)

h ˆ�ni(t)i

Ø  Results SF

Time

i�site

h ˆ�ni(t) ˆ�n0(0)i

Ø  Results SF

Time

i�site

i�site

Re

Im

Ø  Results MI

N�(t)

|⇢eg(t)|

TimeTime

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

THANK YOU FOR YOUR ATTENTION!

Non Markovian Quantum Dynamics Cortona, 26 August 2015