control of quantum two-level systemsย ยท changes ๐ on bloch sphere oscillating fields...
TRANSCRIPT
Control of quantum two-level systems
AS-Chap. 10 - 2
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How to control a qubit?
10.3 Control of quantum two-level systems โฆ.. General concept
Does the answer depend on specific realization? No, only its implementation
Qubit is pseudo spin General concept exists Independent of qubit realization Methods from nuclear magnetic resonance (NMR)
Nuclear magnetic resonance Method to explore the magnetism of nuclear spins Important application Magnetic resonance imaging (MRI) in medicine MRI exploits the different nuclear magnetic signatures of different tissues
AS-Chap. 10 - 3
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10.3 Control of quantum two-level systems โฆ.. General concept
early suggestions H. Carr (1950) and V. Ivanov (1960)
Brief MRI history
1972 MRI imaging machine proposed by R. Damadian (SUNY)
1973 1st MRI image by P. Lauterbur (Urbana-Champaign)
Late 1970ies Fast scanning technique proposed by P. Mansfield (Nottingham)
2003 Nobel Prize in Medicine for P. Lauterbur and P. Mansfield
AS-Chap. 10 - 4
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10.3 Control of quantum two-level systems โฆ.. NMR techniques
Overview: Important NMR techniques
Basic idea Rotate spins by static or oscillating magnetic fields
Static fields parallel to quantization axis free precession changes ๐ on Bloch sphere Oscillating fields perpendicular to quantization axis change population changes ๐ on Bloch sphere
Important protocols Rabi population oscillations Relaxation measurement ๐1 Ramsey fringes ๐2
โ Spin echo (Hanh echo) ๐2
โ corrected for reversible dephasing
AS-Chap. 10 - 5
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10.3. Control of quantum two-level systems โฆ.. Quantum two-level system
Arbitrary TLS ๐ป =1
2
๐ป11 ๐ป12๐ป21 ๐ป22
is Hermitian matrix
๐ป11, ๐ป22 โ โ and ๐ป21 = ๐ป12โ
we choose ๐ป11 > ๐ป22
Hamiltonian of a quantum two-level system (TLS)
Symmetrize Subtract global energy offset ๐ป11+๐ป22
4ร 1
๐ป =1
2
๐ ฮ โ ๐ฮ
ฮ + ๐ฮ โ๐=
๐
2๐ ๐ง +
ฮ
2๐ ๐ฆ +
ฮ
2๐ ๐ฅ
๐ โก๐ป11 โ ๐ป22
2> 0 ฮ, ฮ โ โ
Natural or physical basis ๐+ , |๐โโฉ
๐ป 0 =1
2
๐ 00 โ๐
=๐
2๐ ๐ง
๐ป ๐ยฑ = ยฑ๐
2
AS-Chap. 10 - 6
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10.3 Control of quantum two-level systems โฆ.. Quantum two-level system
Diagonalization of ๐ฏ ๐ป =
1
2
๐ ฮ โ ๐ฮ
ฮ + ๐ฮ โ๐
tan ๐ โกฮ2+ฮ 2
๐ with 0 โค ๐ โค ๐
tan๐ โกฮ
ฮ with 0 โค ๐ โค 2๐
Eigenvalues det๐ โ 2๐ธยฑ ฮ โ ๐ฮ
ฮ + ๐ฮ โ๐ โ 2๐ธยฑ= 0
๐ โ 2๐ธยฑ โ๐ โ 2๐ธยฑ โ ฮ โ ๐ฮ ฮ + ๐ฮ = 0
๐ธยฑ = ยฑ1
2๐2 + ฮ2 + ฮ 2 โก ยฑ
1
2โ๐q
Eigenvectors |ฮจ+โฉ = +๐โ๐๐ 2 cos๐
2๐+ + ๐+๐๐ 2 sin
๐
2|๐โโฉ
|ฮจโโฉ = โ๐โ๐๐ 2 sin๐
2๐+ + ๐+๐๐ 2 cos
๐
2|๐โโฉ
๐ป ฮจยฑ = Eยฑ|ฮจยฑโฉ
C. Cohen-Tannoudji, B. Diu, and F. Laloรซ, Quantum Mechanics Volume One, Wiley-VCH
Basis ฮจ+ , |ฮจโโฉ energy eigenbasis (because ๐ป has energy units)
transition energy
conveniently expressed in
frequency units
AS-Chap. 10 - 7
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10.3 Control of quantum two-level systems โฆ.. Quantum two-level system
Visualization on Bloch sphere
tan ๐ โกฮ2+ฮ 2
๐ with 0 โค ๐ โค ๐
tan๐ โกฮ
ฮ with 0 โค ๐ โค 2๐
|๐น+โฉ = +๐โ๐๐ 2 cos๐
2๐+ + ๐๐๐ 2 sin
๐
2|๐โโฉ
Basis rotation on Bloch sphere!
๐
๐
|๐โโฉ
|๐+โฉ
|๐ณ+โฉ
|๐ณโโฉ
๐
๐ฝ
Vectors |๐นโโฉ and |๐น+โฉ define new quantization axis
๐ป =๐2+ฮ2+ฮ 2
2๐ ๐ง in basis ๐นโ , |๐น+โฉ
|๐นโโฉ and |๐น+โฉ become new poles
|๐นโโฉ = โ๐โ๐๐ 2 sin๐
2๐+ + ๐๐๐ 2 cos
๐
2|๐โโฉ
๐
AS-Chap. 10 - 8
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10.3 Control of quantum two-level systems โฆ.. Qubit Control: Fictitious spin ยฝ
Fictitious spin ยฝ in fictitious magnetic field ๐ฉ
๐ป โ = โ๐พโ ๐ฉ โ ๐บ = โ๐พโ
2
๐ต๐ง ๐ต๐ฅ โ ๐๐ต๐ฆ๐ต๐ฅ + ๐๐ต๐ฆ โ๐ต๐ง
๐พ is the gyromagnetic ratio
๐ต = ๐ต๐ฅ , ๐ต๐ฆ , ๐ต๐ง๐
is the magnetic field vector
Analogy to spin ยฝ in static magnetic field
Fictitous spin in fictitious ๐ฉ-field quantum TLS โ ๐+ โ ๐โ โ ๐ ฮจ+ โ ๐ ฮจโ (๐ denotes the quantization axis along which ๐ป โ is diagonal) โ ๐พ ๐ฉ E+ โ Eโ = โ๐๐
Polar angles of ๐ฉ ๐, ๐ โ๐พโ๐ต๐ง ๐ โ๐พโ๐ต๐ฅ ฮ โ๐พโ๐ต๐ฆ ฮ
Any quantum TLS has a โbuilt-
inโ static magnetic field
โ NMR situation!
Orientation with respect to the quantization axis
depends on ๐, ฮ, ฮ
AS-Chap. 10 - 9
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Free precession
Free precession
free evolution corresponds to free
precession about the ๐-axis
Also called Larmor precession
In absence of decoherence, only ๐ ๐ evolves linearly with time ๐ก
In the energy eigenbasis g , |eโฉ , the qubit state vector |๐น0โฉ is parallel to built-in field for ๐น0 = |gโฉ antiparallel to the built-in field for ๐น0 = |eโฉ Built-in field points along ๐ง-axis on Bloch sphere
๐
๐
|๐ โฉ
|๐โฉ
๐๐
|๐ณ๐โฉ
When qubit state vector |๐น0โฉ is not parallel or antiparallel to built-in field
|๐ณ(๐)โฉ
๐(๐)
๐น = cos๐
2e + ๐๐๐ sin
๐
2g
๐ฝ๐
๐
AS-Chap. 10 - 10
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Free precession
Larmor precession โ formal calcualtion
Energy eigenbasis ๐ป =โ๐๐
2๐ ๐ง fictitious field aligned along quantization axis
๐
๐
|๐ โฉ
|๐โฉ
๐๐
๐ฝ๐
|๐ณ๐โฉ
|๐ณ(๐)โฉ
๐(๐)
Time-independent problem Develop into stationary states
๐น ๐ก = e ๐น0 ๐โ๐๐q๐ก/2|eโฉ + g ๐น0 ๐๐๐q๐ก/2|gโฉ
๐น0 = cos๐02
e + ๐๐๐0 sin๐02
g
๐น ๐ก = ๐โ๐๐q๐ก/2 cos๐02|eโฉ + ๐๐๐0๐๐๐q๐ก/2 sin
๐02|gโฉ
= cos๐02|eโฉ + ๐๐(๐0+๐q๐ก) sin
๐02|gโฉ
๐(๐)
๐น = cos๐
2e + ๐๐๐ sin
๐
2g
๐
AS-Chap. 10 - 11
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10.3 Control of quantum two-level systems โฆ.. Qubit Control: Rabi Oscillations
Rotating drive field
Consider the qubit state vector |๐นโฉ expressed in energy eigenbasis g , |eโฉ
Apply a drive field with amplitude โ๐d rotating about the ๐-axis at frequency ๐
๐ป ๐ก =โ
2
๐q ๐d๐โ๐๐๐ก
๐d๐๐๐๐ก โ๐q
๐
๐
๐ป โ = โ๐พโ
2
๐ต๐ง ๐ต๐ฅ โ ๐๐ต๐ฆ๐ต๐ฅ + ๐๐ต๐ฆ โ๐ต๐ง
๐ฉ๐ = โ๐๐ช
๐๐ ๐ฉ๐ ๐ฉ๐
0 ๐d 0
๐ 4 ๐d 2 ๐d 2
๐ 2 0 ๐d
3๐ 4 โ๐d 2 ๐d 2
๐ โ๐d 0
5๐ 4 โ๐d 2 โ๐d 2
3๐ 2 0 โ๐d
7๐ 4 ๐d 2 โ๐d 2
๐
AS-Chap. 10 - 12
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10.3 Control of quantum two-level systems โฆ.. Qubit Control: Rabi Oscillations
Driven quantum TLS
๐ป ๐ก =โ
2
๐q ๐d๐โ๐๐๐ก
๐d๐๐๐๐ก โ๐q
=โ๐q
2๐ ๐ง +
โ๐d
2๐ +๐
+๐๐๐ก + ๐ โ๐โ๐๐๐ก
Drive rotating about arbitrary equatorial axis on the Bloch sphere
๐ป d =โ๐d
2๐ +๐
+๐ ๐๐ก+๐ + ๐ โ๐โ๐ ๐๐ก+๐
without loss of generality we choose ๐ = 0
โก ๐ป 0 โก ๐ป d
Operators ๐ โ โก e g and ๐ + โก e g create or annihilate an excitation in the TLS
Here, our physics convention is not very intuitive, but consistent!
๐+ =0 01 0
๐โ =0 10 0
Matrix representation and
AS-Chap. 10 - 13
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10.3 Control of quantum two-level systems โฆ.. Qubit Control: Rabi Oscillations
Time evolution of driven quantum TLS ๐ป ๐ก =
โ
2
๐q ๐d๐โ๐๐๐ก
๐d๐๐๐๐ก โ๐q
Qubit state ๐น ๐ก = ๐e ๐ก e + ๐g ๐ก g
obeys Schrรถdinger equation
๐d
d๐ก๐e ๐ก =
๐q
2๐e(๐ก) +
๐d
2๐โ๐๐๐ก๐g(๐ก)
๐d
d๐ก๐g ๐ก =
๐d
2๐๐๐๐ก๐e(๐ก) โ
๐q
2๐g(๐ก)
๐โd
d๐ก๐น ๐ก = ๐ป ๐น ๐ก
Time-dependent Difficult to solve Move to rotating frame!
Schrรถdinger equation looses explicit time dependence
๐d
d๐ก๐e ๐ก =
๐q โ ๐
2๐e(๐ก) +
๐d
2๐g(๐ก)
๐d
d๐ก๐g ๐ก =
๐d
2๐e(๐ก) โ
๐q โ ๐
2๐g(๐ก)
๐e ๐ก โก ๐๐๐๐ก 2 ๐e ๐ก
๐g ๐ก โก ๐โ๐๐๐ก 2 ๐g ๐ก
AS-Chap. 10 - 14
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Rabi Oscillations
Formal treatment Effective Hamiltonian ๐ป describing the same dynamics as ๐ป (๐ก)
๐โd
d๐ก๐น ๐ก = ๐ป ๐น ๐ก
๐น ๐ก โก ๐e ๐ก e + ๐g ๐ก g
๐d
d๐ก๐e ๐ก =
๐q โ ๐
2๐e(๐ก) +
๐d
2๐g(๐ก)
๐d
d๐ก๐g ๐ก =
๐d
2๐e(๐ก) โ
๐q โ ๐
2๐g(๐ก)
Interpretation of the rotating frame ๐ป ๐ก =
โ
2
๐q ๐d๐โ๐๐๐ก
๐d๐๐๐๐ก โ๐q
The frame rotates at the angular speed ๐ of the drive Driving field appears at rest Drive can be in resonance with Larmor precession frequency ๐q
Away from resonance |๐ โ ๐q| โซ 0
red terms dominate no ๐ โ |๐โฉ transitions induced by drive Near resonance ๐ โ ๐q
blue terms dominate ๐ โ |๐โฉ transitions induced by drive
H =โ
2
โฮ๐ ๐d
๐d ฮ๐
with ฮ๐ โก ๐ โ ๐q
AS-Chap. 10 - 15
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Rabi Oscillations
Expand into stationary states
Dynamics of the effective Hamiltonian
H =โ
2
โฮ๐ ๐d
๐d ฮ๐
ฮ๐ โก ๐ โ ๐q
Diagonalize H =โโ ฮ๐2 + ๐d
2
2๐ ๐ง
Initial state ๐ณ๐ = ๐ (energy ground state)
๐น ๐ก = ๐น โ ๐น 0 ๐+๐๐ก2 ฮ๐2+๐d
2
๐น โ + ๐น + ๐น 0 ๐โ๐๐ก2 ฮ๐2+๐d
2
๐น +
tan ๐ = โ๐d
ฮ๐
tan๐ = 0
|๐น +โฉ = + cos๐
2e + sin
๐
2|gโฉ
|๐น โโฉ = โ sin๐
2e + cos
๐
2|gโฉ
|ฮจ+โฉ = +๐โ๐๐ 2 cos๐
2๐+ + ๐+๐๐ 2 sin
๐
2|๐โโฉ
|ฮจโโฉ = โ๐โ๐๐ 2 sin๐
2๐+ + ๐+๐๐ 2 cos
๐
2|๐โโฉ
New eigenstates
๐น ๐ก = cos๐
2๐+๐๐ก2 ฮ๐2+๐d
2
๐น โ + sin๐
2๐โ๐๐ก2 ฮ๐2+๐d
2
๐น +
AS-Chap. 10 - 16
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Rabi Oscillations
Probability ๐ท๐ to find TLS in |๐โฉ
ฮ๐ โก ๐ โ ๐q
๐e โก e ๐น ๐ก 2 = e ๐น ๐ก2=
|๐น +โฉ = + cos๐
2e + sin
๐
2|gโฉ
|๐น โโฉ = โ sin๐
2e + cos
๐
2|gโฉ
= sin๐
2cos
๐
2๐โ๐๐ก2 ฮ๐2+๐d
2
โ ๐+๐๐ก2 ฮ๐2+๐d
22
๐น ๐ก = cos๐
2๐+๐๐ก2
ฮ๐2+๐d2
๐น โ + sin๐
2๐โ๐๐ก2
ฮ๐2+๐d2
๐น +
= sin ๐ sin๐ก ฮ๐2 +๐d
2
2
2
๐e =๐d2
ฮ๐2 + ๐d2 sin
2๐ก ฮ๐2 + ๐d
2
2
tan ๐ = โ๐d
ฮ๐
Driven Rabi oscillations TLS population oscillates under transversal drive
AS-Chap. 10 - 17
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Rabi Oscillations
Rabi Oscillations on the Bloch sphere
ฮ๐ โก ๐ โ ๐q
๐น ๐ก = cos๐
2๐+๐๐ก2
ฮ๐2+๐d2
๐น โ + sin๐
2๐โ๐๐ก2
ฮ๐2+๐d2
๐น +
๐e =๐d2
ฮ๐2 + ๐d2 sin
2๐ก ฮ๐2 + ๐d
2
2
tan ๐ = โ๐d
ฮ๐
On resonance ๐ = ๐๐ช
Rotating frame cancels Larmor precession
State vector ๐น ๐ก has no ๐-evolution
๐น ๐ก = cos๐d๐ก
2g + ๐ sin
๐d๐ก
2e
Rotation about ๐ฅ-axis
๐
๐
๐ณ ๐ = |๐ โฉ
|๐ณ (๐)โฉ
๐ ๐๐๐
Finite detuning ๐ซ๐ > ๐ Additional precession at ๐ฅ๐
Population oscillates faster Reduced oscillation amplitude
AS-Chap. 10 - 18
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Rabi Oscillations
Rabi Oscillations โ Graphical representation ๐e =
๐d2
ฮ๐2 + ๐d2 sin
2๐ก ฮ๐2 + ๐d
2
2
On resonance ๐ = ๐๐ช
Detuning ๐ซ๐ = ๐๐๐
First experimental demonstration with superconducting qubit Y. Nakamura et al., Nature 398, 786 (1999)
๐ท๐
๐
๐
๐ท๐
๐
๐
๐. ๐
AS-Chap. 10 - 19
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Rabi Oscillations
Oscillating vs. rotating drive โ Microwave pulses ๐e =
๐d2
ฮ๐2 + ๐d2 sin
2๐ก ฮ๐2 + ๐d
2
2
On resonance (๐ = ๐๐ช) ๐ท๐
๐
๐
2๐/๐
2๐d
Oscillating drive 2โ๐d cos๐๐ก = โ๐d ๐+i๐๐ก + ๐โi๐๐ก
in frame rotating with +๐ the ๐โi๐๐ก -component rotates fast with โ2๐ For ๐d โช ๐ this fast contribtion averages out on the timescale of the slowly rotating component Rotating wave approximation
AS-Chap. 10 - 20
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Rabi Oscillations
Important drive pulses on the Bloch sphere
๐
๐
๐
๐
๐
๐
|๐ โฉ
๐
๐ -pulse (๐dฮ๐ก = ๐) g โ |eโฉ flips, refocus phase evolution
๐ /๐-pulse (๐dฮ๐ก = ๐/2) Rotates into equatorial plane and back
๐
๐ โ ๐ ๐
๐
2๐/๐
2โ๐d
ฮ๐ก
๐ + ๐ ๐
๐
AS-Chap. 10 - 21
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Rabi Oscillations
Rabi Oscillations in presence of decoherence ๐e =
๐d2
ฮ๐2 + ๐d2 sin
2๐ก ฮ๐2 + ๐d
2
2
๐ซ๐ =0 ๐ท๐
๐
๐
Effect of decoherence (qualitative approach) Loss of coherent properties to environment at a constant rate ฮdec = 2๐/๐dec โChange of coherenceโ (time derivative) proportional to โamount of coherenceโ
Exponential decay of coherent property with factor ๐โ๐ช๐๐๐๐
๐๐ = ๐โ
๐
๐ป๐๐๐ Argument holds well for population decay (energy relaxation, ๐1) Loss of phase coherence more diverse depending on environment (exponential, Gaussian, or power law) Experimental timescales range from few ns to 100 ฮผs
๐. ๐ ๐. ๐ ๐ + ๐โ๐
๐ป๐๐๐
Decay envelope
AS-Chap. 10 - 22
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Rabi Oscillations
Rabi decay time ๐e =
๐d2
ฮ๐2 + ๐d2 sin
2๐ก ฮ๐2 + ๐d
2
2
๐ซ๐ =0 ๐ท๐
๐
๐
๐. ๐ ๐. ๐ ๐ + ๐โ๐
๐ป๐๐๐
Complicated interplay between ๐1, ๐2โ, and the drive
At long times, small oscillations persist Nevertheless useful order-of-magnitude check for ๐dec Important tool for single-qubit gates To determine ๐1, ๐2
โ, ๐2 correctly, more sophisticated protocols are required energy relaxation measurements, Ramsey fringes, spin echo
AS-Chap. 10 - 23
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Rabi Oscillations
Energy relaxation on the Bloch sphere
๐
๐
|๐ โฉ
๐
๐ฝ
๐ณ ๐ฝ,๐
๐
Environment induces energy loss
State vector collapses to g Implies also loss of phase information Intrinsically irreversible
๐1-time rate ฮ1 =2๐
๐1
Golden Rule argument
ฮ1 โ ๐ ๐q
๐ ๐ is noise spectral density High frequency noise Intuition: Noise induces transitions
Quantum jumps
Single-shot, quantum nondemotiton measurement yields a discrete jump to g at a random time
Probability is equal for each point of time
Exponential decay with ๐โ
๐ก
๐1
AS-Chap. 10 - 24
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Rabi Oscillations
Dephasing on the Bloch sphere Environment induces random phase changes
ฮ2 โ ๐ ๐ โ 0 , ๐2 =2๐
ฮ2
Low-frequency noise is detuned No energy transfer
1/๐-noise ๐ ๐ โ1
๐
Example: Two-level fluctuator bath To some extent reversible
Decay laws ๐โ
๐ก
๐2, ๐โ
๐ก
๐2
2 ,
๐ก
๐2
๐ฝ
๐
๐
|๐ โฉ
|๐โฉ
๐๐ ๐
๐ณ ๐ฝ๐, ๐๐
|๐ณ(๐)โฉ
๐
๐
๐ฝ๐
Visualization Phase ๐ becomes
more and more unknown with time
Classical probility, no superposition!
Phase coherence lost when arrows are distributed over whole equatorial plane
AS-Chap. 10 - 25
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Rotating frame & no detuning (ฮ๐ = ๐ โ ๐q = 0) โ no ๐ฅ๐ฆ-evolution
x
y 1
0
x
y 1
0
x
y 1
0
x
y 1
0
x
y 1
0
Qubit dynamics โ Relaxation
10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Energy Relaxation
๐ซ๐
๐ Measurement Init
๐ป๐
๐ท๐
๐ซ๐
๐
๐โ๐
AS-Chap. 10 - 26
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x
y y y y
free evolution time
x
y 1
0
x
y 1
0
x
y 1
0
10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Ramsey fringes
Qubit dynamics โ Ramsey fringes (๐ป๐โ )
x
y
๐ซ๐๐๐
๐ /๐ Measurement Init
๐ /๐
๐ซ๐๐๐ ๐ป๐โ
๐ซ๐ = ๐
๐ท๐
๐
๐. ๐ ๐. ๐ ๐ + ๐โ๐
๐ซ๐ > ๐ Beating ฮ๐ = 2๐/ฮ๐ ๐ป๐
โ โ๐ = ๐๐ป๐โ๐ + ๐ป๐
โ๐ Energy decay always present!
AS-Chap. 10 - 27
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free evolution time
x
y 1
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x
y 1
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Spin echo
Qubit dynamics โ Spin echo (๐ป๐๐ฌโ )
๐ซ๐๐๐/๐
๐ /๐ Measurement Init
๐ /๐
๐ซ๐๐๐ ๐ป๐๐ฌโ
๐ซ๐ = ๐ ๐ท๐
๐. ๐ ๐. ๐ ๐ โ ๐โ๐
๐
Refocussing time
๐ซ๐๐๐/๐
x
y
x x x x
y
x x x x
y x
y 1
0
Refocussing pulse reverses low-frequency phase dynamics ๐ป๐๐ฌ
โ > ๐ป๐โ
AS-Chap. 10 - 28
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10.3 Control of quantum two-level systems
โฆ.. Qubit Control: Ramsey vs. Spin echo sequence
Ramsey vs. spin echo sequence
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
F(ฯ
t/2)
ฯt/2
Ramsey filter FR
spin echo filter FE
Spin echo cancels the effect of low-frequency noise in the environment
Pulse sequences act as filters! Environment described by noise spectral density ๐(๐)
๐ญ๐ ๐๐ก =sin2
๐๐ก2
๐๐ก2
2
๐ญ๐ ๐๐ก =sin4
๐๐ก4
๐๐ก4
2
Decay envelope โ ๐โ๐ก2 S ๐ ๐นR,E
๐๐ก
2๐๐
+โโโ
Spin echo sequence
Filters low-frequency noise for ๐๐ก โ 0 ๐๐ก โ 2 Noise field fluctuates synchronously with ๐-pulse No effect
Sequence length ๐ก is important!