Quantum Simulation of the
Haldane Phase
19.12.2013
HUJI
Alex Retzker
Sussex
Quantum Simulations with Trapped Ions, 2013
Itsik Cohen
Accepted to PRL
BOSCH UND SIEMENS HAUSGERÄTE GRUPPE
MAGIC - Magnetic Gradient Induced Coupling
V
z
1
F. Mintert and C Wunderlich,
PRL 87, 257904 (2001);
0
B
MAGIC - Magnetic Gradient Induced Coupling
MAGIC
δz
|1
Use microwave
instead of laser light
F. Mintert and C Wunderlich,
PRL 87, 257904 (2001);
|0
B
Short Qubit coherence time
Ramsey experiment
T ≈ 5 ms
F=1
F=0
mF=-1
mF=0
mF=0
mF=+1
Hahn Echo:
Sussex I 19.12.2013 I Folie: 5
Background
Carr Purcell – CP
Spin echo decay
A sequence of echos, i.e., of π pulses
focuses the polarization for a long time
z
y
x
Sussex I 19.12.2013 I Folie: 6
Carr Purcell – CP:
Spin echo decay
A sequence of echos, i.e., of π pulses
focuses the polarization for a long time
z
z
y
x
π+δΦ
2 δΦ
y
x
Sussex I 19.12.2013 I Folie: 7
Composite pulses
z
é q ù
Goal: Rf (q ) = exp ê -i s f ú
ë 2 û
é q (1+ e ) ù
Real pulse: Mf (q ) = exp ê -i
sf ú
2
ë
û
but: Mf (2kp ) = ±R(2kpe )
y
x
And now we can use the Suzuki
Trotter decomposition
The optimization is on operations not on memory but
theoretically the difference is very small.
Torosov & Vitanov, PRA 87,
043418 (2013). Kyoseva &
Vitanov arxiv:1310.7145.
Wang et al., arxiv: 1312.4523
Kenneth R. Brown, Aram W. Harrow, and Isaac L. Chuang, PRA 70, 052318 (2004)
Sussex I 19.12.2013 I Folie: 8
Coherent control
Timoney et, al., 2007
Montangero et,. Al. PRL
99, 170501 (2007)
Sussex I 19.12.2013 I Folie: 9
Search for a stable qubit
0
No dephasing
but no coupling

Can we somehow
construct two ‘good’
qubit levels?
1
0'
1
Coupled but strongly
dephased
Sussex I 19.12.2013 I Folie: 10
Dynamical Decoupling: take I
B
1
w0
D
1
Dephasing(T2) Rate:
Flipping(T1) Rate:
Dephasing(T2) Rate:
+second order
B effects
Sussex I 19.12.2013 I Folie: 11
Dynamical Decoupling: take II
0
Flipping(T1) Rate:
+ Relative phase fluctuations
+1
u
0'
-1
Dephasing(T2) Rate:
D
0'
+second order
B effects
d
Sussex I 19.12.2013 I Folie: 12
Ramsey measurement results
N. Timoney, I. Baumgart, M. Johanning, A. F. Varon, M. B. Plenio, A.
Retzker & Ch. Wunderlich. Nature 476 (2011)
Sussex I 19.12.2013 I Folie: 13
Rabi Oscillation of the Sussex group
S. C. Webster, S. Weidt, K. Lake, J. J. McLoughlin, and
W. K. Hensinger. PRL 111, 140501 (2013)
Sussex I 19.12.2013 I Folie: 14
Generalisation to N levels
General conditions:
Dj J Z Di = 0
for each i,j
Robustness to
external noise
H d Di = 0
for each i
Robustness to
control noise
Level
structure
of the
calcium
ion.
N. Aharon, M. Drewsen, and A. Retzker, PRL 111, 230507 (2013)
Sussex I 19.12.2013 I Folie: 15
Generalisation to N levels
N. Aharon, M. Drewsen, and A. Retzker, PRL 111, 230507 (2013)
Sussex I 19.12.2013 I Folie: 16
The Boulder Scheme
C. Ospelkaus, et. al., PRL 101, 090502 (2008)
C. Ospelkaus, et. al., Nature 476, 181 (2011)
Oxford group
D.P.L Aude Craik, et al., arxiv: 1308.2078
Magnetometry
locking the signal to the frequency of the
pulses(Rabi frequency)
Kotler et al., Nature, 473 (2011)
Magnetometry
locking to the frequency and not the Rabi frequency
I. Baumgart, J.-M. Cai, A. Retzker, M. Plenio, and Ch. Wunderlich, In preparation
Magnetometry
dWg » 0.47Hz / Hz
I. Baumgart, J.-M. Cai, A. Retzker, M. Plenio, and Ch. Wunderlich, In preparation
The Haldane Phase in the S=1 XXZ Antiferromagnetic chain
H = åS S + S S + lS S + D ( S
i i+1
x x
i i+1
y y
i i+1
z z
)
i 2
z
l>0
i
Flip flops
Neel Order
Invariant under global rotations around z and global spin flips
a
Ostring
= lim i- j ®¥ s ija
j-1
æ
ö a
a
a
a
s ij = -Si Exp çip å Sl ÷ S j
è l=i+1 ø
The Haldane Phase in the S=1 XXZ Antiferromagnetic chain
H = åS S + S S + lS S + D ( S
i i+1
x x
i i+1
y y
i i+1
z z
)
i 2
z
i
Finite energy gap, short range correlations. (Haldane, 1983)
Nonlocal string order parameter (Tasaki and Kennedy, 1987)
Symmetry protected double-degeneracy of the entanglement spectrum (Pollmann et at.,
2010)
Spin degrees of freedom
u
0
D
0'
+1
0'
d
-1
This setup only kills external
magnetic noise, but is not
robust to power fluctuations
The decoherence
free subspace:
We have to work
in a decoherence
free subspace
DD , ud , du
Sussex I 19.12.2013 I Folie: 24
The
H = å Sxi Sxi+1 + Syi Syi+1
term
i
u
0
+1
u
u
0'
-1
D
d
Analogous to a red/blue sidband interaction
Flip flops will happend automatically if we
start in the DFS
Sussex I 19.12.2013 I Folie: 25
Two-qubit gate
almost:
Gets into a fully
entangled state in
the middle;
Schmidt number 3
The effective Hamiltonian – single qubit
u
u
u
D
u
D
For zero
temperature
Sz - S
2
z
d
d
Has no effect
For a thermal
state
+
b bSz
Has no effect
Sussex I 19.12.2013 I Folie: 27
The effective Hamiltonian – two qubit
u
u
u
u
D
d
Virtual phonon
u
u
D
d
Sussex I 19.12.2013 I Folie: 28
The effective Hamiltonian – the D term
H = D (S
)
i 2
z
W
0 0
Dr
2
r
0
Dr
0 =( u - d )/ 2
Wr
+1
0'
-1
W
W 2
»
u u + d d )=
Sz
(
2D r
2D r
2
r
2
r
The effective Hamiltonian – the λ term
H1 = a S S =
i j
x x
H = lS S
i i+1
z z
a i i+1 i i+1 i i+1 i i+1
S+ S- + S+ S- + S+ S+ + S- S- )
(
4
By adding a term of the form:
H 2 = w Sz
a i i+1 i i+1 i i+1 i i+1
H1 = ( S+ S- + S+ S- + S+ S+ + S- S- ) =
4
a i i+1 i i+1
= ( Sx Sx + Sy Sy )
2
The effective Hamiltonian – the λ term
H = lS S
H1 = a ( S S + S S
i j
x x
i j
y y
By adding a term of the form:
H 2 = w ( cosJ Sz -sin J Sy )
z
x
y
)
i i+1
z z
The effective Hamiltonian – the λ term
0
Wy
d
d
H 2 = W y Sy
In the
interaction
picture
Wy
+1
0'
-1
H = lS S
i i+1
z z
Reaching the Haldane phase
All the transitions are
second order and thus
hard to cross
To break the
symmetries we add the
term:
H = -hå(-1) S
i
i
i
z
Detecting the Haldane phase
1) String order:
2) Double degenrate entanglement spectrum
3) Gap and exponentialy decaying correlation function
Thanks a lot for your attention!
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