quantum simulation of the haldane phase
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Quantum Simulation of the Haldane Phase. 19.12.2013 HUJI Alex Retzker Sussex Quantum Simulations with Trapped Ions, 2013 Itsik Cohen Accepted to PRL. MAGIC - Magnetic Gradient Induced Coupling. V. z. F. Mintert and C Wunderlich, PRL 87 , 257904 (2001); . - PowerPoint PPT PresentationTRANSCRIPT
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B O S C H U N D S I E M E N S H A U S G E R Ä T E G R U P P E
Quantum Simulation of the Haldane Phase
19.12.2013HUJI
Alex Retzker
SussexQuantum Simulations with Trapped Ions, 2013
Itsik Cohen
Accepted to PRL
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z
V
MAGIC - Magnetic Gradient Induced Coupling
F. Mintert and C Wunderlich, PRL 87, 257904 (2001);
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MAGIC - Magnetic Gradient Induced Coupling
MAGIC
F. Mintert and C Wunderlich, PRL 87, 257904 (2001); |0
|1δ z
Use microwave instead of laser light
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Short Qubit coherence time
Ramsey experiment
T ≈ 5 ms
F=1
F=0
mF=0mF=-1
mF=+1
mF=0
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Hahn Echo:
Sussex I 19.12.2013 I Folie: 5
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Background
Spin echo decay
Sussex I 19.12.2013 I Folie: 6
Carr Purcell – CP
A sequence of echos, i.e., of π pulses focuses the polarization for a long time
y
z
x
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Spin echo decay
Sussex I 19.12.2013 I Folie: 7
Carr Purcell – CP:
A sequence of echos, i.e., of π pulses focuses the polarization for a long time
y
z
xπ+δΦ
y
z
x
2 δΦ
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Sussex I 19.12.2013 I Folie: 8
Composite pulses
Goal:
y
z
x
Real pulse:
but:
And now we can use the Suzuki Trotter decomposition
The optimization is on operations not on memory but theoretically the difference is very small.
Kenneth R. Brown, Aram W. Harrow, and Isaac L. Chuang, PRA 70, 052318 (2004)
Torosov & Vitanov, PRA 87, 043418 (2013). Kyoseva & Vitanov arxiv:1310.7145.Wang et al., arxiv: 1312.4523
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Sussex I 19.12.2013 I Folie: 9
Coherent control
Timoney et, al., 2007
Montangero et,. Al. PRL 99, 170501
(2007)
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Sussex I 19.12.2013 I Folie: 10
Search for a stable qubit
1
1
0
0'
No dephasing but no
coupling
Coupled but strongly dephased
Can we somehow construct two
‘good’ qubit levels?
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Sussex I 19.12.2013 I Folie: 11
Dynamical Decoupling: take I
1
1Dephasing(T2)
Rate:
Flipping(T1) Rate:Dephasing(T2)
Rate: +second order B effects
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Sussex I 19.12.2013 I Folie: 12
Dynamical Decoupling: take II
Dephasing(T2) Rate: +second
order B effects
Flipping(T1) Rate:
+Relative phase fluctuations
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Ramsey measurement results
Sussex I 19.12.2013 I Folie: 13
N. Timoney, I. Baumgart, M. Johanning, A. F. Varon, M. B. Plenio, A. Retzker & Ch. Wunderlich. Nature 476 (2011)
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Rabi Oscillation of the Sussex group
Sussex I 19.12.2013 I Folie: 14
S. C. Webster, S. Weidt, K. Lake, J. J. McLoughlin, and W. K. Hensinger. PRL 111, 140501 (2013)
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Generalisation to N levels
Sussex I 19.12.2013 I Folie: 15
General conditions:
for each i,j
for each i
Robustness to external noiseRobustness to control noise
N. Aharon, M. Drewsen, and A. Retzker, PRL 111, 230507 (2013)
Level structure
of the calcium
ion.
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Generalisation to N levels
Sussex I 19.12.2013 I Folie: 16
N. Aharon, M. Drewsen, and A. Retzker, PRL 111, 230507 (2013)
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The Boulder Scheme
C. Ospelkaus, et. al., PRL 101, 090502 (2008)C. Ospelkaus, et. al., Nature 476, 181 (2011)
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D.P.L Aude Craik, et al., arxiv: 1308.2078
Oxford group
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Magnetometry locking the signal to the frequency of
the pulses(Rabi frequency)
Kotler et al., Nature, 473 (2011)
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Magnetometry
locking to the frequency and not the Rabi frequency
I. Baumgart, J.-M. Cai, A. Retzker, M. Plenio, and Ch. Wunderlich, In preparation
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Magnetometry
I. Baumgart, J.-M. Cai, A. Retzker, M. Plenio, and Ch. Wunderlich, In preparation
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The Haldane Phase in the S=1 XXZ Antiferromagnetic chain
Flip flops
Invariant under global rotations around z and global spin flips
Neel Order
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The Haldane Phase in the S=1 XXZ Antiferromagnetic chain
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)
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Spin degrees of freedom
Sussex I 19.12.2013 I Folie: 24
This setup only kills external magnetic noise,
but is not robust to power fluctuations
We have to work in a
decoherence free subspace
The decoherence
free subspace:
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The
Sussex I 19.12.2013 I Folie: 25
Analogous to a red/blue sidband interaction
Flip flops will happend automatically if we start in the DFS
term
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Two-qubit gate
almost:
Gets into a fully entangled state in the middle;
Schmidt number 3
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The effective Hamiltonian – single qubit
Sussex I 19.12.2013 I Folie: 27
Has no effect
For zero temperature
For a thermal state Has no
effect
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The effective Hamiltonian – two qubit
Sussex I 19.12.2013 I Folie: 28
Virtual phonon
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The effective Hamiltonian – the D term
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The effective Hamiltonian – the λ term
By adding a term of the form:
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The effective Hamiltonian – the λ term
By adding a term of the form:
x y
z
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The effective Hamiltonian – the λ term
In the interaction
picture
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Reaching the Haldane phase
All the transitions are second order and thus hard to
crossTo break the
symmetries we add the term:
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Detecting the Haldane phase
1) String order:
2) Double degenrate entanglement spectrum
3) Gap and exponentialy decaying correlation function
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B O S C H U N D S I E M E N S H A U S G E R Ä T E G R U P P E
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