Quantum Phase Diagram of Bosons in Optical Lattices
Quantum Phase Diagram of Bosons in Optical Lattices
Quantum Phase Diagram of Bosons in Optical Lattices
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<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong><br />
<strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong><br />
Ednilson Santos<br />
Freie Universität Berl<strong>in</strong><br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 1
Outl<strong>in</strong>e <strong>of</strong> the talk<br />
1. Experimental facts<br />
2. Second-quantized Hamiltonian<br />
3. Mean-field theory<br />
4. Corrections to mean-field theory<br />
5. Second method<br />
6. Perturbation theory<br />
7. Results<br />
8. Perspectives<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 2
1 - Experimental facts<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 3
2 - Second-quantized Hamiltonian<br />
∫<br />
Ĥ =<br />
{<br />
d 3 x<br />
[ −¯h<br />
ˆψ † 2<br />
]<br />
(x)<br />
2m ∇2 + V (x) − µ<br />
}<br />
ˆψ (x) + 2πa¯h2<br />
m ˆψ † (x) 2 ˆψ (x)<br />
2<br />
Wannier decomposition: ˆψ(x) =<br />
∑i âiw(x − x i )<br />
Bose-Hubbard Hamiltonian:<br />
[ U<br />
2 ˆn i(ˆn i − 1) − µˆn i<br />
]<br />
Here: ˆn i = â † iâi<br />
Ĥ BH = −t ∑ 〈i,j〉<br />
â † iâj + ∑ i<br />
Matrix elements:<br />
⎧<br />
⎨t = − ∫ d 3 xw ∗ (x)<br />
⎩U = 4πa¯h2<br />
m<br />
{<br />
}<br />
− ¯h2<br />
2m ▽2 +V (x) w(x)<br />
∫<br />
d 3 x|w(x)| 4 <strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 4
3 - Mean-field theory<br />
∑<br />
〈i,j〉<br />
â † iâj → 2D ∑ i<br />
(ψ ∗ â i + ψâ † i − |ψ|2 )<br />
Local Hamiltonian:<br />
Ĥ MF = ∑ [<br />
−2Dt<br />
(ψ ∗ â i + ψâ † i − |ψ|2) + U ]<br />
2 ˆn i(ˆn i − 1) − µˆn i<br />
i<br />
Partition function: Z = Tr<br />
Self-consistency relations:<br />
⎧<br />
⎨∂F MF<br />
∂ψ = 0<br />
⎩∂F MF<br />
∂ψ<br />
= 0 ∗<br />
[ ]<br />
e −βĤMF(ψ∗ ,ψ)<br />
=⇒<br />
⎧<br />
⎨〈â † i 〉 = ψ∗<br />
⎩〈â i 〉 = ψ<br />
= e −βF MF(ψ ∗ ,ψ)<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 5
4 - Corrections to mean-field theory<br />
Introduction <strong>of</strong> smallness parameter:<br />
]<br />
Ĥ (η, ψ ∗ , ψ) = ĤMF (ψ ∗ , ψ) + η<br />
[ĤBH − ĤMF (ψ ∗ , ψ)<br />
Partition function: Z = Tr<br />
Extremization <strong>of</strong> free energy:<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
∂F<br />
∂ψ = 0<br />
∂F<br />
∂ψ ∗ = 0<br />
[ ]<br />
e −βĤ(η,ψ∗ ,ψ)<br />
= e −βF(η,ψ∗ ,ψ)<br />
=⇒ Interpretation with<strong>in</strong> VPT<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 6
4.1 - Procedure<br />
1. Expansion <strong>in</strong> η def<strong>in</strong>es level <strong>of</strong> approximation<br />
Zeroth order: Mean-field phase diagram<br />
First order: No shift <strong>of</strong> phase boundary<br />
Second order: <strong>Quantum</strong> correction to phase boundary<br />
2. Landau expansion:<br />
[<br />
]<br />
F (N) (η, ψ ∗ , ψ) = N S a (N)<br />
0 (η) + a (N)<br />
2 (η)|ψ| 2 + a (N)<br />
4 (η)|ψ| 4 + · · ·<br />
Here a (N)<br />
2p (η) is truncated expansion <strong>in</strong> η up to order N <strong>of</strong> a 2p(η):<br />
⎧<br />
⎨a 2 (η) = (2Dt) 2 (1 − η) 2 ∑ ∞<br />
m=0 (−tη)m α (m)<br />
2 + 2Dt(1 − η)<br />
⎩<br />
a 2p (η) = (2Dt) 2p (1 − η) 2p ∑ ∞<br />
m=0 (−tη)m α (m)<br />
2p ; p ≠ 1<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 7
4.2 - <strong>Phase</strong> diagram<br />
Second-order phase transition ⇐⇒ a (N)<br />
4 (η = 1) > 0<br />
<strong>Phase</strong> boundary: a (N)<br />
2 (η = 1) = 0<br />
General structure:<br />
⎧<br />
⎨a (0)<br />
2 (η = 1) = α(0) 2 t2 + 2Dt<br />
[<br />
]<br />
⎩a (N)<br />
2 (η = 1) = (−1) N t 2 α (N)<br />
2 t N + α (N−1)<br />
2 t N−1<br />
; N ≥ 1<br />
<strong>Phase</strong> boundary:<br />
⎧⎪⎨<br />
⎪ ⎩<br />
t (0)<br />
c<br />
t (N)<br />
c<br />
= − 2D<br />
α (0)<br />
2<br />
= − α(N−1) 2<br />
α (N)<br />
2<br />
; N ≥ 1<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 8
4.3 - Explicit formulas<br />
⎧<br />
α (0)<br />
2 = 1+b<br />
µ(b−1)<br />
⎪⎨<br />
α (1)<br />
2 = 2D(1+b)2<br />
µ 2 (b−1) 2<br />
⎪⎩<br />
α (2)<br />
2 = 4D [D(b−3)(b+1) 3 −3b(b−3+4b 2 −2b 3 )]<br />
Here: n = 1 and b = µ/U<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
(b−3)(b−1) 3 µ 3<br />
t (0)<br />
c<br />
U = (1−b)b<br />
2D(b+1)<br />
t (1)<br />
c<br />
U = (1−b)b<br />
2D(b+1)<br />
t (2)<br />
c<br />
U = b(b−3)(b−1)(b+1) 2<br />
6b(b−3+4b 2 −2b 3 )−2D(b−3)(b+1) 3<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 9
5 - Second method<br />
Bose-Hubbard Hamiltonian with current:<br />
Ĥ BH (J ∗ , J) = ĤBH + ∑ ( )<br />
J ∗ â i + Jâ † i<br />
i<br />
ψ = 〈â i 〉 = 1 ∂F(J ∗ , J)<br />
N s ∂J ∗ ; ψ ∗ = 〈â † i 〉 = 1 ∂F(J ∗ , J)<br />
N s ∂J<br />
Legendre transformation: Γ(ψ ∗ , ψ) = ψ ∗ J + ψJ ∗ − F/N s<br />
∂Γ<br />
∂ψ ∗ = J ;<br />
Physical limit <strong>of</strong> vanish<strong>in</strong>g current:<br />
∂Γ<br />
∂ψ ∗ = 0 ;<br />
∂Γ<br />
∂ψ = J ∗<br />
∂Γ<br />
∂ψ = 0 <strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 10
5.1 - Details<br />
F(J ∗ , J) = F 0 +<br />
∞∑<br />
c 2n |J| 2n<br />
n=1<br />
Γ(ψ ∗ , ψ) = −F 0 + 1 |ψ| 2 − 3 c 4<br />
c 2 c 4 |ψ| 4 + · · ·<br />
2<br />
with: c 2p = ∑ ∞<br />
n=0 (−t)n α (n)<br />
2p<br />
Remark: α (n)<br />
2p<br />
are the same used for the first method.<br />
<strong>Phase</strong> boundary:<br />
⎧<br />
1<br />
= 1 ⎨<br />
c 2 α (0) ⎩ 1 + α(1) 2<br />
2 α (0)<br />
2<br />
t +<br />
⎡(<br />
⎣<br />
α (1)<br />
2<br />
α (0)<br />
2<br />
) 2<br />
− α(2) 2<br />
α (0)<br />
2<br />
⎤ ⎫<br />
⎬<br />
⎦t 2 + · · ·<br />
⎭ = 0<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 11
5.2 - <strong>Phase</strong> boundary<br />
First order: t (1)<br />
c = − α(0) 2<br />
α (1)<br />
2<br />
Remark: Identical to mean-field phase boundary.<br />
Second order:<br />
t (2)<br />
c =<br />
α 1<br />
2 ( )<br />
α 2 − α 2 ±<br />
1<br />
1<br />
2 ( α 2 − α 2 1<br />
√<br />
) α 2 1 − 4 ( α 2 1 − α )<br />
2<br />
with: α 1 = α(1) 2<br />
α (0)<br />
2<br />
; α 2 = α(2) 2<br />
α (0)<br />
2<br />
Note: Choose the smallest critical t (2)<br />
c .<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 12
6 - Perturbation theory<br />
here Ĥ = Ĥ0 + λˆV .<br />
Ĥ|Ψ n 〉 = E n |Ψ n 〉<br />
Perturbative series <strong>in</strong> λ :<br />
∞∑<br />
|Ψ n 〉 = λ k |Ψ (k)<br />
n 〉 ; E n =<br />
k=0<br />
∞∑<br />
k=0<br />
λ k E (k)<br />
n<br />
with: E (k)<br />
n = 〈Ψ (0)<br />
n |ˆV |Ψ (k−1)<br />
n 〉<br />
|Ψ (k)<br />
n 〉 = ∑ |Ψ (0)<br />
m≠n<br />
m 〉 〈Ψ(0)<br />
E (0)<br />
m |ˆV |Ψ (k−1)<br />
n 〉<br />
n − E (0)<br />
m<br />
−<br />
k∑<br />
l=1<br />
E (l)<br />
n<br />
∑<br />
|Ψ (0)<br />
m 〉 〈Ψ(0) m |Ψ n (k−l) 〉<br />
m≠n E n (0) − E m<br />
(0)<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 13
6.1 - <strong>Diagram</strong>matic notation<br />
E (1)<br />
n = 〈Ψ (0)<br />
n |ˆV |Ψ (0)<br />
n 〉 =<br />
E n (2) = ∑ m≠n<br />
E n (3) = ∑<br />
1<br />
E n (0) − E m<br />
(0)<br />
∑<br />
m 1 ≠n m 2 ≠n<br />
〈Ψ (0)<br />
n |ˆV |Ψ (0)<br />
m 〉〈Ψ (0)<br />
m |ˆV |Ψ (0)<br />
n 〉 =<br />
1<br />
E (0)<br />
n − E (0)<br />
1<br />
m 1<br />
E (0)<br />
n − E (0)<br />
m 2<br />
〈Ψ (0)<br />
×〈Ψ (0)<br />
m 2<br />
|ˆV |Ψ (0)<br />
m 1<br />
〉〈Ψ (0)<br />
m 1<br />
|ˆV |Ψ (0)<br />
n 〉 − ∑<br />
m 1 ≠n<br />
m 2<br />
〉<br />
n |ˆV |Ψ (0)<br />
×〈Ψ (0)<br />
n |ˆV |Ψ (0)<br />
n 〉〈Ψ (0)<br />
n |ˆV |Ψ (0)<br />
m 1<br />
〉〈Ψ (0)<br />
m 1<br />
|ˆV |Ψ (0)<br />
n 〉<br />
= −<br />
1<br />
(E (0)<br />
n − E (0)<br />
m 1<br />
) 2<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 14
6.2 - Rules<br />
Each dot represents an <strong>in</strong>teraction ˆV .<br />
The <strong>in</strong>ternal l<strong>in</strong>es appear<strong>in</strong>g between two consecutive dots<br />
” q |Ψ (0)<br />
m 〉〈Ψ (0)<br />
m |,<br />
are associated with the factor ∑ m≠n<br />
“ 1<br />
E m (0) −E n<br />
(0)<br />
and q is the number <strong>of</strong> l<strong>in</strong>es l<strong>in</strong>k<strong>in</strong>g the two given<br />
consecutive dots.<br />
If there is more than one l<strong>in</strong>e between two consecutive dots,<br />
each extra l<strong>in</strong>e is associated with an extra disconnected part<br />
<strong>of</strong> the diagram.<br />
The external l<strong>in</strong>es are associated with the unperturbed bra<br />
and ket 〈Ψ (0)<br />
n | · · · |Ψ (0)<br />
n 〉.<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 15
6.3 - More <strong>in</strong>teraction terms<br />
Example:<br />
Ĥ = Ĥ0 + λˆV + σŴ<br />
⎧<br />
⎨ˆV →<br />
⎩Ŵ →<br />
E n = E (0)<br />
n + λ + σ + λσ<br />
+λ 2 + σ 2 + · · ·<br />
(<br />
+<br />
)<br />
Symmetries between diagrams if ˆV and<br />
Ŵ are Hermitean:<br />
= ; =<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 16
6.4 - Lattice calculations<br />
First approach:<br />
(ψ ∗ â i + ψâ † i − |ψ|2)<br />
Ĥ(η, ψ ∗ , ψ) = −tη ∑ â † iâj − 2Dt(1 − η) ∑<br />
〈i,j〉<br />
i<br />
[ ]<br />
U<br />
2 ˆn i(ˆn i − 1) − µˆn i<br />
+ ∑ i<br />
Second approach:<br />
Ĥ BH (J ∗ , J) = −t ∑ 〈i,j〉<br />
â † iâj+ ∑ i<br />
( )<br />
J ∗ â i + Jâ † i<br />
+ ∑ i<br />
[ U<br />
2 ˆn i(ˆn i − 1) − µˆn i<br />
]<br />
Note: Both Hamiltonians are essentially equivalent.<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 17
6.5 - Interaction terms <strong>in</strong> our case<br />
Ĥ BH (J ∗ , J) = −t ∑ 〈i,j〉<br />
â † iâj+ ∑ i<br />
( )<br />
J ∗ â i + Jâ † i<br />
+ ∑ i<br />
[ U<br />
2 ˆn i(ˆn i − 1) − µˆn i<br />
]<br />
⎧∑<br />
⎪⎨<br />
∑<br />
⎪⎩ ∑<br />
i âi →<br />
i ↠i →<br />
〈i,j〉 ↠iâj →<br />
Ground-state energy:<br />
E 0 = E (0)<br />
0 + t 2 (<br />
( + |J| 2 +<br />
+|J| 2 t + +<br />
+ + +<br />
)<br />
)<br />
+ · · ·<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 18
6.6 - Individual processes<br />
The previous diagrams can be split <strong>in</strong> their <strong>in</strong>dividual processes<br />
which are represented by arrow diagrams. For example:<br />
= 2 1 ; = 1 2<br />
= 1 2<br />
3<br />
= 1<br />
+<br />
2 3<br />
4 3<br />
2 1<br />
4 +<br />
+<br />
2 3<br />
4 1<br />
2 4<br />
3 1<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 19
6.7 - Evaluation<br />
g =<br />
4<br />
3<br />
2<br />
1<br />
= N s (n + 1)(n + 2)<br />
1<br />
(E (0)<br />
n − E (0)<br />
1<br />
n+1 )2 E (0)<br />
n − E (0)<br />
n+2<br />
1. Create a particle at some site with N s different possibilities:<br />
g = N s<br />
h √n + 1/(E<br />
(0)<br />
n − E (0)<br />
n+1 ) i<br />
· · ·<br />
2. Create a second particle at the same position <strong>of</strong> the first:<br />
h √n i h<br />
(0)<br />
g = N s + 1/(E n − E (0) √n i<br />
n+1 ) (0) + 2/(E n − E (0)<br />
n+2 ) · · ·<br />
3. Annihilate one particle at the site <strong>of</strong> the two previously<br />
created particles:<br />
g = N s<br />
h √n<br />
+ 1/(E<br />
(0)<br />
n − E (0)<br />
n+1 ) i h √n<br />
+ 2/(E<br />
(0)<br />
n − E (0)<br />
n+2 ) i h √n<br />
+ 2/(E<br />
(0)<br />
n − E (0)<br />
n+1 ) i<br />
· ·<br />
4. Annihilate the last particle:<br />
g = N s<br />
h √n<br />
+ 1/(E<br />
(0)<br />
n − E (0)<br />
n+1 ) i h √n<br />
+ 2/(En − E (0)<br />
n+2 ) i h √n<br />
+ 2/(E<br />
(0)<br />
n − E (0)<br />
n+1 ) i √n<br />
+ 1<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 20
6.8 - Coefficients<br />
α (0)<br />
2 = 2 1 + 1 2<br />
α (1)<br />
2 = 1 2<br />
3 + 5 permutations<br />
α (2)<br />
2 = 1<br />
2<br />
3<br />
4 + 23 permutations<br />
+<br />
+<br />
2 3<br />
4 1<br />
2 4<br />
3 1<br />
+ 15 permutations<br />
+ 7 permutations<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 21
6.9 - Explicit results<br />
α (0)<br />
2 =<br />
b + 1<br />
U(b − n)(b + 1 − n)<br />
α (1)<br />
2 =<br />
2D(b + 1) 2<br />
U 2 (b − n) 2 (b + 1 − n) 2<br />
α (2)<br />
2 = 2D { 2D(b + 1) 3 (b − 2 − n)(b + 3 − n) + n(b − n)(b + 1 − n)<br />
×(1 + n)(4 + 3b + 2n) [ −3 − 2n + 2(b 2 + b − 2bn + n 2 ) ]}<br />
/ [ U 3 (b − n − 2)(b − n) 3 (b + 1 − n) 3 (b + 3 − n) ]<br />
Here n is the number <strong>of</strong> particles at each site and b = µ/U .<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 22
7 - Results<br />
First Method:<br />
Second method:<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 23
8 - Perspectives<br />
Automatization for higher orders<br />
Resummation<br />
Extension for T ≠ 0<br />
<strong>Phase</strong> boundary from diverg<strong>in</strong>g Green’s function<br />
<strong>Quantum</strong> <strong>Phase</strong> <strong>Diagram</strong> <strong>of</strong> <strong>Bosons</strong> <strong>in</strong> <strong>Optical</strong> <strong>Lattices</strong> – p. 24