Download Partial collapse and uncollapse of a wavefunction: theory and

UCR, 5/06/10
Partial collapse and uncollapse of a
wavefunction: theory and experiments
(what is “inside” collapse)
Alexander Korotkov
EE dept. (& Physics), University of California, Riverside
Outline: • Bayesian formalism for quantum measurement
• Persistent Rabi oscillations (+ expt.)
• Wavefunction uncollapse (+ expts.)
Acknowledgements:
Theory: R. Ruskov, A. Jordan, K. Keane
Expt.: UCSB (J. Martinis, N. Katz et al.),
Saclay (D. Esteve, P. Bertet et al.)
Alexander Korotkov
Funding:
University of California, Riverside
Quantum mechanics is weird…
Niels Bohr:
“If you are not confused by
quantum physics then you
haven’t really understood it”
Richard Feynman:
“I think I can safely say that nobody
understands quantum mechanics”
Weirdest part is quantum measurement
Alexander Korotkov
University of California, Riverside
Quantum mechanics =
Schrödinger equation (evolution)
+
collapse postulate (measurement)
1) Probability of measurement result pr =| 〈ψ
2) Wavefunction after measurement =
ψr
|ψ r 〉 |
• State collapse follows from common sense
• Does not follow from Schrödinger Eq. (contradicts)
What is “inside” collapse?
What if collapse is stopped half-way?
Alexander Korotkov
2
University of California, Riverside
What is the evolution due to measurement?
(What is “inside” collapse?)
• controversial for last 80 years, many wrong answers, many correct answers
• solid-state systems are more natural to answer this question
Various approaches to non-projective (weak, continuous,
partial, generalized, etc.) quantum measurements
Names: Davies, Kraus, Holevo, Mensky, Caves, Knight, Plenio, Walls,
Carmichael, Milburn, Wiseman, Gisin, Percival, Belavkin, etc.
(very incomplete list)
Key words: POVM, restricted path integral, quantum trajectories, quantum
filtering, quantum jumps, stochastic master equation, etc.
Our limited scope:
(simplest system,
experimental setups)
Alexander Korotkov
solid-state qubit
detector
I(t), noise S
University of California, Riverside
“Typical” setup: double-quantum-dot (DQD)
qubit + quantum point contact (QPC) detector
|1Ò
|2Ò
H
|1Ò
|2Ò
e
Gurvitz, 1997
|2Ò
|1Ò
I(t)
I(t)
H = HQB + HDET + HINT
ε
H QB = σ Z + Hσ X
2
ΔI
I (t ) = I0 +
z(t ) + ξ (t )
2
const + signal + noise
Two levels of average detector current: I1 for qubit state |1〉, I2 for |2〉
Response: ΔI = I1 – I2
Detector noise: white, spectral density SI
For low-transparency QPC
H DET = ∑ l El al†al + ∑ r Er ar†ar + ∑ l ,r T ( ar†al + al†ar )
H INT = ∑ l ,r ΔT ( c1†c1 − c2†c2 ) ( ar†al + al†ar )
5/38
Alexander Korotkov
S I = 2eI
University of California, Riverside
Bayesian formalism for DQD-QPC system
Qubit evolution due to measurement (quantum back-action):
ψ ( t ) = α ( t ) | 1〉 + β ( t ) | 2〉 or ρ ij ( t )
HQB = 0
|1Ò
H
|2Ò
1) |α(t)|2 and |β(t)|2 evolve as probabilities,
i.e. according to the Bayes rule (same for ρii)
e
I(t)
2) phases of α(t) and β(t) do not change
(no dephasing!), ρij /(ρii ρjj)1/2 = const
Bayes rule (1763, Laplace-1812):
prior
probab. likelihood
posterior
probability
P ( Ai | res) =
P ( Ai ) P (res | Ai )
∑ k P ( Ak ) P (res | Ak )
Alexander Korotkov
1 τ
I ( t ) dt
∫
0
τ
I1
I2
(A.K., 1998)
measured
So simple because:
1) no entaglement at large QPC voltage
2) QPC happens to be an ideal detector
3) no Hamiltonian evolution of the qubit
University of California, Riverside
Bayesian formalism for a single qubit
H
Vg V
e
I(t)
I(t) 2e
•
•
• Time derivative of the quantum Bayes rule
• Add unitary evolution of the qubit
• Add decoherence γ (if any)
ρ 11 = - ρ 22 = -2 H Im ρ12 + ρ11 ρ 22
•
ρ 12
2ΔI
[ I (t ) - I0 ]
SI
ΔI
= i ερ12 + i H ( ρ11 - ρ 22 ) + ρ12 ( ρ11 - ρ 22 ) [ I ( t ) - I 0 ] - γ ρ12
SI
γ = Γ - ( ΔI ) 2 / 4 S I , Γ − ensemble decoherence
(A.K., 1998)
γ = 0 for QPC detector
Averaging over result I(t) leads to conventional master equation with Γ
Evolution of qubit wavefunction can be monitored if γ=0 (quantum-limited)
Natural generalizations: • add classical back-action
• entangled qubits
Alexander Korotkov
University of California, Riverside
Assumptions needed for the Bayesian formalism:
• Detector voltage is much larger than the qubit energies involved
eV >> ÑΩ, eV >> ÑΓ, Ñ/eV << (1/Ω, 1/Γ), Ω=(4H 2+ε2)1/2
(no coherence in the detector, classical output, Markovian approximation)
• Simpler if weak response, |ΔI | << I0, (coupling C ~ Γ/Ω is arbitrary)
Derivations:
1) “logical”: via correspondence principle and comparison with
decoherence approach (A.K., 1998)
2) “microscopic”: Schr. eq. + collapse of the detector (A.K., 2000)
qubit
ρ ijn ( t )
detector
quantum
interaction
n ( tk )
pointer
frequent
quantum collapse
classical
information
n – number of electrons
passed through detector
3) from “quantum trajectory” formalism developed for quantum optics
(Goan-Milburn, 2001; also: Wiseman, Sun, Oxtoby, etc.)
4) from POVM formalism (Jordan-A.K., 2006)
5) from Keldysh formalism (Wei-Nazarov, 2007)
Alexander Korotkov
University of California, Riverside
Why not just use Schrödinger
equation for the whole system?
qubit
detector
information
Impossible in principle!
Technical reason: Outgoing information (measurement result)
makes it an open system
Philosophical reason: Random measurement result, but
deterministic Schrödinger equation
Einstein: God does not play dice
Heisenberg: unavoidable quantum-classical boundary
Alexander Korotkov
University of California, Riverside
Quantum limit for ensemble decoherence
Γ = (ΔI)2/4SI + γ
ensemble
decoherence rate
“measurement time” (S/N=1)
single-qubit
decoherence
~ information flow [bit/s]
γ ≥ 0 ⇒ Γ ≥ (ΔI)2/4SI
( ΔI ) 2 / 4 S I
η = 1- =
Γ
Γ
γ
τ m = 2 SI /( ΔI ) 2
Γτ m ≥
1
2
A.K., 1998, 2000
S. Pilgram et al., 2002
A. Clerk et al., 2002
D. Averin, 2000,2003
detector ideality (quantum efficiency)
η ≤ 100%
Translated into energy sensitivity: (εO εBA)1/2 ≥ =/2
εO, εBA: sensitivities [J/Hz] limited by output noise and back-action
Danilov, Likharev,
Zorin, 1983
Known since 1980s (Caves, Clarke, Likharev, Zorin, Vorontsov, Khalili, etc.)
(εO εBA - εO,BA2)1/2 ≥ =/2 ⇔ Γ ≥ (ΔI)2/4SI + K2SI/4
Quantum limits for measurement are due to quantum (informational) back-action
10/38
Alexander Korotkov
University of California, Riverside
POVM vs. Bayesian formalism
General quantum measurement (POVM formalism) (Nielsen-Chuang, p. 85,100):
†
M
ρ
M
M rψ
Measurement (Kraus) operator
r
r
→
ρ
or
†
Mr (any linear operator in H.S.) : ψ →
|| M rψ ||
Probability : Pr = || M r ψ ||2
Completeness :
†
M
∑ r r Mr = 1
or
Tr( M r ρ M r )
Pr = Tr( M r ρ M r† )
(People often prefer linear evolution
and non-normalized states)
• POVM is essentially a projective measurement in an extended Hilbert space
• Easy to derive: interaction with ancilla + projective measurement of ancilla
• For extra decoherence: incoherent sum over subsets of results
Relation between POVM and
quantum Bayesian formalism:
decomposition
M r = U r M r† M r
unitary Bayes
Mathematically POVM and quantum Bayes are almost equivalent
We focus not on the mathematical structures, but on
particular setups and experimental consequences
Alexander Korotkov
University of California, Riverside
Can we verify the Bayesian formalism
experimentally?
Direct way:
prepare
partial
measur.
control
(rotation)
projective
measur.
A.K.,1998
However, difficult: bandwidth, control, efficiency
(expt. realized only for supercond. phase qubits)
Tricks are needed for real experiments
(proposals 1999-2010)
Alexander Korotkov
University of California, Riverside
Non-decaying (persistent) Rabi oscillations
excited
left
right
ground
z
(measure left-right)
to verify:
stop & check
z
|e〉 •
- Relaxes to the ground state if left alone (low-T)
- Becomes fully mixed if coupled to a high-T
(non-equilibrium) environment
- Oscillates persistently between left and right
if (weakly) measured continuously
ρ11
|left〉
•
1.0
0.5
•|g〉
Reρ11
|right〉
Imρ11
0.0
-0.5
0
5
Direct experiment is difficult
Alexander Korotkov
10
15
time
20
25
30
A.K., PRB-1999
University of California, Riverside
Indirect experiment: spectrum
of persistent Rabi oscillations A.K., LT’1999
qubit
C
detector
I(t)
12
(const + signal + noise)
C=13
SI(ω)/S0
10
10
6
coordinate
amplifier noise fl higher pedestal,
1
poor quantum efficiency,
ηÜ1
but the peak is the same!!!
0
0
1
C = ( ΔI ) 2 / HS I
ω/Ω
4
integral under the peak ‹ variance ‚z2Ú
3
1
0.3
2
0
z is Bloch
SI (ω)
Ω = 2H
8
ΔI
I (t ) = I0 +
z(t ) + ξ (t )
2
A.K.-Averin, 2000
0.0
0.5
1.0
1.5
ω /Ω
Ω - Rabi frequency
2.0
peak-to-pedestal ratio = 4η ≤ 4
Ω 2 ( ΔI ) 2 Γ
S I (ω ) = S0 + 2
(ω − Ω 2 ) 2 + Γ 2ω 2
Alexander Korotkov
How to distinguish experimentally
persistent from non-persistent? Easy!
perfect Rabi oscillations: ·z2Ò=·cos2Ò=1/2
imperfect (non-persistent): ·z2Ò Ü 1/2
quantum (Bayesian) result: ·z2Ò = 1 (!!!)
(demonstrated in Saclay expt.)
University of California, Riverside
2
z
|e〉 •
|left〉
•
•|g〉
|right〉
How to understand ·z2Ò = 1?
ΔI
I (t ) = I0 +
z(t ) + ξ (t )
2
First way (mathematical)
We actually measure operator:
z → σz
z2 → σz2 = 1
qubit
detector
I(t)
(What does it mean?
Difficult to say…)
Second way (Bayesian)
ΔI 2
ΔI
S I (ω ) = Sξξ +
S zz (ω ) +
Sξ z (ω )
4
2
quantum back-action changes z
in accordance with the noise ξ
Equal contributions (for weak
coupling and η=1)
“what you see is what you get”: observation becomes reality
Can we explain it in a more reasonable way (without spooks/ghosts)?
+1
z(t)?
No (under assumptions of macrorealism;
Leggett-Garg, 1985)
-1
or some other z(t)?
15/38
Alexander Korotkov
University of California, Riverside
Leggett-Garg-type inequalities for
continuous measurement of a qubit
Ruskov-A.K.-Mizel, PRL-2006
Jordan-A.K.-Büttiker, PRL-2006
I(t)
Assumptions of macrorealism
(similar to Leggett-Garg’85):
I ( t ) = I 0 + ( ΔI / 2) z ( t ) + ξ ( t )
| z ( t ) | ≤ 1, 〈ξ ( t ) z( t + τ )〉 = 0
Leggett-Garg,1985
Kij = ·Qi Qj Ò
if Q = ±1, then
1+K12+K23+K13≥0
K12+K23+K34 -K14 £2
Then for correlation function
K (τ ) = 〈 I ( t ) I ( t + τ )〉
K (τ 1 ) + K (τ 2 ) − K (τ 1 + τ 2 ) ≤ ( ΔI / 2)
2
6
4
2
0
SI (ω)
≤ 4S0
detector
SI(ω)/S0
qubit
1 ω/Ω 2
0
quantum result
violation
3
( ΔI / 2) 2
2
3
×
2
and for area under narrow spectral peak
2
2
[
S
(
f
)
−
S
]
df
≤
(8
/
π
)
(
Δ
I
/
2)
0
∫ I
η is not important!
( ΔI / 2) 2
×
8
Experimentally measurable violation
(Saclay experiment)
Alexander Korotkov
π2
University of California, Riverside
Experiment with supercond. qubit (Saclay group)
〈 n〉 = 0.23
photons
〈 n〉 = 3.9
photons
frequency (MHz)
LGI violation
• superconducting charge qubit
0.75
2/3
0.50
0.25
0.00
0
Alexander Korotkov
A. Palacios-Laloy, F. Mallet,
F. Nguyen, P. Bertet, D. Vion,
D. Esteve, and A. Korotkov,
Nature Phys., 2010
1
(transmon) in circuit QED setup
8/π2
1.00
peak area
spectral density (z)
courtesy of
Patrice Bertet
(this fig.
unpub.)
5 10 15 20 25
f (MHz)
(microwave reflection from cavity)
• driven Rabi oscillations
• perfect spectral peaks
• LGI violation (both K & S)
University of California, Riverside
Next step: quantum feedback (Useful?)
Goal: persistent Rabi oscillations with zero linewidth (synchronized)
Types of quantum feedback:
Direct
a la Wiseman-Milburn
(1993)
(monitor quantum state
and control deviation)
desired evolution
feedback
signal comparison
control stage
circuit
(barrier height)
ρij(t)
Bayesian
equations
I(t)
H
e
Δ H fb / H = F × Δ ϕ
D (feedback fidelity)
environment
1.00
0.95
Cenv /Cdet= 0
0.90
0.1 0.5
C=Cdet=1
τa=0
0.85
D (feedback fidelity)
C<<1 detector
qubit
2
3
4
5
6
7
8
F (feedback strength)
9
10
Ruskov & A.K., 2002
Alexander Korotkov
= F × φm
fb
H
control
× cos (Ω t), τ-average
C<<1
detector
I(t)
1.0
0.8
0.6
averaging time
τ a = (2π/Ω)/10
0.4
C=1
η =1
0.2
0.0
0.0
0.2
F (f
db
0.4
k t
0.6
th)
0.8
Ruskov & A.K., 2002
1.0
ηeff = 1
Y
φm
C = 0.1
τ [(ΔI)2/SI] = 1
0.8
0.5
0
0.5
ε/H0= 1
0.6
0
0.1
Ω =0.2
Δ Ω /C
0.4
0.2
0.2
X
local oscil.
× sin (Ω t), τ-average
0.80
1
(do as in usual classical
feedback)
ΔH
qubit
F (feedback strength)
0
Imperfect but simple
(apply measurement signal to
control with minimal processing)
ΔH fb / H = F sin( Ωt )
⎛ I ( t ) − I0
⎞
×⎜
− cos Ωt ⎟
⎝ ΔI / 2
⎠
“Simple”
rel. phase
Best but very difficult
D (feedback fidelity)
Bayesian
0.1
0.0
0.0
0.2
0.4
0.6
F/C (feedback strength)
A.K., 2005
University of California, Riverside
0.8
Quantum feedback in optics
First experiment: Science 304, 270 (2004)
First detailed theory:
H.M. Wiseman and G. J. Milburn,
Phys. Rev. Lett. 70, 548 (1993)
Alexander Korotkov
University of California, Riverside
Quantum feedback in optics
First experiment: Science 304, 270 (2004)
n
w
a
r
d
h
t
i
rw
e
p
pa
PRL 94, 203002 (2005) also withdrawn
First detailed theory:
H.M. Wiseman and G. J. Milburn,
Phys. Rev. Lett. 70, 548 (1993)
20/38
Alexander Korotkov
Recent experiment:
Cook, Martin, Geremia,
Nature 446, 774 (2007)
(coherent state discrimination)
University of California, Riverside
Undoing a weak measurement of a qubit
(“uncollapse”) A.K. & Jordan, PRL-2006
It is impossible to undo “orthodox” quantum
measurement (for an unknown initial state)
Is it possible to undo partial quantum measurement?
(To restore a “precious” qubit accidentally measured)
Yes! (but with a finite probability)
If undoing is successful, an unknown state is fully restored
weak (partial)
(unknown) measurement
ψ0
ψ1
(partially
collapsed)
ful
s
s
ψ0 (still
e
c
c
u
s
unknown)
unsucces
sful
uncollapse
(information erasure)
Alexander Korotkov
ψ2
University of California, Riverside
Quantum erasers in optics
Quantum eraser proposal by Scully and Drühl, PRA (1982)
Interference fringes restored for two-detector
correlations (since “which-path” information
is erased)
Our idea of uncollapsing is quite different:
we really extract quantum information and then erase it
Alexander Korotkov
University of California, Riverside
Uncollapse of a qubit state
Evolution due to partial (weak, continuous, etc.) measurement is
non-unitary, so impossible to undo it by Hamiltonian dynamics.
How to undo? One more measurement!
| 1〉
| 1〉
×
| 0〉
| 1〉
=
| 0〉
| 0〉
need ideal (quantum-limited) detector
(similar to Koashi-Ueda, PRL-1999)
Alexander Korotkov
(Figure partially adopted from
Jordan-A.K.-Büttiker, PRL-06)
University of California, Riverside
Uncollapsing for DQD-QPC system
A.K. & Jordan, 2006
first (“accidental”) uncollapsing
measurement measurement
H =0
Qubit
(DQD)
I(t)
r(t)
r0
Detector
(QPC)
t
Simple strategy: continue measuring
until result r(t) becomes zero! Then
any initial state is fully restored.
(same for an entangled qubit)
However, if r = 0 never happens, then
uncollapsing procedure is unsuccessful.
Probability of success:
Alexander Korotkov
Meas. result:
ΔI t
r(t ) =
[ ∫ I ( t ') dt ' - I 0 t ]
SI 0
If r = 0, then no information
and no evolution!
e
-|r0|
PS = |r |
-|r |
e 0 ρ11 (0) + e 0 ρ 22 (0)
University of California, Riverside
General theory of uncollapsing
POVM formalism
(Nielsen-Chuang, p.100)
Measurement operator Mr :
Probability : Pr = Tr( M r
Uncollapsing operator:
max(C ) = min i
Probability of success:
ρ M r† )
ρ→
Completeness :
C × M r−1
M r ρ M r†
Tr( M r ρ M r† )
†
M
∑ r r Mr = 1
(to satisfy completeness,
eigenvalues cannot be >1)
pi , pi – eigenvalues of M r† M r
min Pr
PS ≤
Pr ( ρ in )
A.K. & Jordan, 2006
Pr(ρin) – probability of result r for initial state ρin,
min Pr – probability of result r minimized over
all possible initial states
Averaged (over r ) probability of success:
Pav ≤ ∑ r min Pr
(cannot depend on initial state, otherwise get information)
25/38
(similar to Koashi-Ueda, 1999)
Alexander Korotkov
University of California, Riverside
Partial collapse of a Josephson phase qubit
|1〉
|0〉
Γ
N. Katz, M. Ansmann, R. Bialczak, E. Lucero,
R. McDermott, M. Neeley, M. Steffen, E. Weig,
A. Cleland, J. Martinis, A. Korotkov, Science-06
How does a qubit state evolve
in time before tunneling event?
(What happens when nothing happens?)
⎧| out 〉 , if tunneled
⎪
ψ = α | 0〉 + β | 1〉 → ψ ( t ) = ⎨ α | 0〉 + β e-Γ t / 2e iϕ | 1〉
, if not tunneled
⎪
2
2 -Γt
α
β
e
|
|
+
|
|
⎩
Main idea:
(better theory: L. Pryadko & A.K., 2007)
amplitude of state |0〉 grows without physical interaction
finite linewidth only after tunneling!
continuous null-result collapse
(similar to optics, Dalibard-Castin-Molmer, PRL-1992)
Alexander Korotkov
University of California, Riverside
Superconducting phase qubit at UCSB
Courtesy of Nadav Katz (UCSB)
X
Iμw
Flux
bias
Idc+Iz
Iμw
Qubit
SQUID
IS
VS
Y 10ns
Z
Iz
Reset
Compute
Meas. Readout
time
Idc
Is
0 1
Vs
|1〉
|0〉
3ns
Repeat 1000x
prob. 0,1
ω01
1 Φ0
Alexander Korotkov
University of California, Riverside
Experimental technique for partial collapse
Nadav Katz et al.
(John Martinis group)
Protocol:
1) State preparation
(via Rabi oscillations)
2) Partial measurement by
lowering barrier for time t
3) State tomography (microwave + full measurement)
Measurement strength
p = 1 - exp(-Γt )
is actually controlled
by Γ, not by t
p = 0: no measurement
p = 1: orthodox collapse
Alexander Korotkov
University of California, Riverside
Experimental tomography data
ψ in =
Nadav Katz et al. (UCSB, 2005)
| 0〉 + | 1〉
2
p=0
p = 0.06
p = 0.14
p = 0.23
p = 0.32
p = 0.43
p = 0.56
p = 0.70
θy
θx
π
π/2
Alexander Korotkov
p = 0.83
University of California, Riverside
Partial collapse: experimental results
Visibility
• In case of no tunneling
phase qubit evolves
lines - theory
dots and squares – expt.
no fitting parameters in (a) and (b)
• Evolution is described
by the Bayesian theory
probability p
without fitting parameters
Azimuthal angle
Polar angle
N. Katz et al., Science-06
• Phase qubit remains
coherent in the process
of continuous collapse
(expt. ~80% raw data,
pulse ampl. ~96% corrected for T1,T2)
in (c) T1=110 ns, T2=80 ns (measured)
quantum efficiency
η0 > 0.8
probability p
30/38
Alexander Korotkov
University of California, Riverside
Uncollapse of a phase qubit state
1) Start with an unknown state
2) Partial measurement of strength
3) π-pulse (exchange |0Ú ↔ |1Ú)
4) One more measurement with
the same strength p
5) π-pulse
A.K. & Jordan, 2006
p
p = 1 - e-Γt
If no tunneling for both measurements,
then initial state is fully restored!
α | 0〉 + β | 1〉 →
α | 0〉 + e iφ β e −Γt / 2 | 1〉
Norm
|1〉
|0〉
Γ
→
e iφα e −Γt / 2 | 0〉 + e iφ β e −Γt / 2 | 1〉
= e iφ (α | 0〉 + β | 1〉 )
Norm
phase is also restored (spin echo)
Alexander Korotkov
University of California, Riverside
Experiment on wavefunction uncollapse
state
preparation
partial
measure p
partial
measure p
π
N. Katz, M. Neeley, M. Ansmann,
R. Bialzak, E. Lucero, A. O’Connell,
H. Wang, A. Cleland, J. Martinis,
and A. Korotkov, PRL-2008
tomography &
final measure
Iμw
p
Idc
10 ns
7 ns
p
10 ns
7 ns
Nature News
Nature-2008
time
Physics
Uncollapse protocol:
- partial collapse
- π-pulse
- partial collapse
(same strength)
ψ in =
Alexander Korotkov
| 0〉+ | 1〉
2
State tomography with
X, Y, and no pulses
Background PB should
be subtracted to find
qubit density matrix
University of California, Riverside
Experimental results on the Bloch sphere
Initial
state
| 1〉
| 0〉 − i | 1〉
2
| 0〉+ | 1〉
2
N. Katz et al.
| 0〉
Partially
collapsed
Uncollapsed
uncollapsing
works well!
Both spin echo (azimuth) and uncollapsing (polar angle)
Difference: spin echo – undoing of an unknown unitary evolution,
uncollapsing – undoing of a known, but non-unitary evolution
Alexander Korotkov
University of California, Riverside
Quantum process tomography
N. Katz et al.
(Martinis group)
p = 0.5
uncollapsing works
with good fidelity!
Why getting worse at p>0.6?
Energy relaxation pr = t/T1= 45ns/450ns = 0.1
Selection affected when 1-p ~ pr
Overall: uncollapsing is well-confirmed experimentally
Alexander Korotkov
University of California, Riverside
Experiment on uncollapsing
using single photons
Kim et al., Opt. Expr.-2009
initial
state
after
partial
collapse
after
uncollapse
• very good fidelity of uncollapsing (>94%)
• measurement fidelity is probably not good
(normalization by coincidence counts)
35/38
Alexander Korotkov
University of California, Riverside
Suppression of T1-decoherence
by uncollapsing
Protocol:
storage period t
(low temperature)
partial collapse
towards ground
state (strength p)
uncollapse
(measurem.
strength pu)
best for 1 - pu = (1 - p ) exp(-t / T1 )
Ideal case (T1 during storage only)
for initial state |ψin〉=α |0〉 +β |1〉
|ψf〉= |ψin〉 with probability (1-p) e-t/T1
|ψf〉= |0〉 with (1-p)2|β|2 e-t/T1(1-e-t/T1)
QPT fidelity (Favs, Fχ)
ρ11
π
A.K. & K. Keane,
PRA-2010
π
1.0
Ideal
0.9
0.8
-t/T1 (1-p)
e
1
p u=
almost
complete
suppression
e-t/T1 = 0.3
p u= p
0.7
without
uncollapsing
0.6
0.5
0.0
0.2
0.4
0.6
0.8
measurement strength p
Trade-off: fidelity vs. probability
procedure preferentially selects
events without energy decay
Alexander Korotkov
1.0
University of California, Riverside
QPT fidelity, probability
Realistic case (T1 and Tϕ at all stages)
1.0
fidelity
0.8
0.6
κ2 = 0.3
}without
uncollapsing
pr ob
0.4
0.2
as in
expt.
(1-pu) κ3κ4 = (1-p) κ1κ2
− ti / T1
abili κκ ϕ = κ1, 0.95
ty = = κ = 1, 0.999,
1
3
κi = e
−t / T
κϕ = e Σ ϕ
0.0
0.0
0.2
0.4
0.6
4
0.99. 0.9
0.8
measurement strength p
1.0
• Tϕ-decoherence is not affected
• fidelity decreases at p→1 due to T1
between 1st π-pulse and 2nd meas.
Trade-off: fidelity vs. selection probability
Alexander Korotkov
• Easy to realize experimentally
(similar to existing experiment)
• Improved fidelity can be observed
with just one partial measurement
Uncollapse seems the only way
to protect against T1-decoherence
without quantum error correction
A.K. & K. Keane,
PRA-2010
University of California, Riverside
Conclusions
● It is easy to see what is “inside” collapse: simple Bayesian
formalism works for many solid-state setups
● Rabi oscillations are persistent if weakly measured
● Collapse can sometimes be undone if we manage
to erase extracted information (uncollapsing)
● Continuous/partial measurements and uncollapsing
may be useful
● Three direct solid-state experiments have been realized,
many interesting experimental proposals are still waiting
38/38
Alexander Korotkov
University of California, Riverside