Wayne State Univ., 05/10/05 Continuous quantum measurement ...
Transcript of Wayne State Univ., 05/10/05 Continuous quantum measurement ...
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University of California, RiversideAlexander Korotkov
Continuous quantum measurementof solid-state qubits and
quantum feedback
Wayne State Univ., 05/10/05
Alexander KorotkovUniversity of California, Riverside
Outline: • Introduction (quantum measurement) • Bayesian formalism for continuous quantum
measurement of a single quantum system• Experimental predictions and proposals
General theme: Information and collapse in quantum mechanics
Acknowledgement: Rusko Ruskov Support:
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University of California, RiversideAlexander Korotkov
Niels Bohr:“If you are not confused byquantum physics then you haven’t really understood it”
Richard Feynman:“I think I can safely say that nobodyunderstands quantum mechanics”
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University of California, RiversideAlexander Korotkov
Quantum mechanics =Schroedinger equation
+collapse postulate
1) Probability of measurement result pr =
2) Wavefunction after measurement =
2| | |rψ ψ⟨ ⟩rψ
• State collapse follows from common sense• Does not follow from Schr. Eq. (contradicts; Schr. cat,
random vs. deterministic)
What if measurement is continuous?(as practically always in solid-state experiments)
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University of California, RiversideAlexander Korotkov
Einstein-Podolsky-Rosen (EPR) paradoxPhys. Rev., 1935
In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system.
1 2 2 1( , ) ( ) ( )n nnx x x u xψ ψ=∑1 2 1 2 1 2( , ) exp[( / ) ( ) ] ~ ( )x x i x x p dp x xψ δ
∞
−∞= − −∫
1x 2x Measurement of particle 1 cannot affect particle 2,while QM says it affects(contradicts causality)
(nowadays we call it entangled state)
=> Quantum mechanics is incomplete
Bohr’s reply (seven pages, one formula: ∆p ∆q ~ h)(Phys. Rev., 1935)It is shown that a certain “criterion of physical reality” formulated … by A. Einstein, B. Podolsky and N. Rosen contains an essential ambiguity when it is applied to quantum phenomena.
Crudely: No need to understand QM, just use the result
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University of California, RiversideAlexander Korotkov
Bell’s inequality (John Bell, 1964)
1 2 1 21 ( )2
ψ = ↑ ↓ − ↓ ↑
Perfect anticorrelation of mea-surement results for the same measurement directions, a b=
a b
(setup due to David Bohm)
Is it possible to explain the QM result assuming local realism and hidden variables or collapse “propagates” instantaneously (faster than light, “spooky action-at-a-distance”)?
( , ) 1, ( , ) 1A a B bλ λ= ± = ± (deterministic result withhidden variable λ)
Assume:
Then: | ( , ) ( , ) | 1 ( , )P a b P a c P b c− ≤ +( ) ( ) ( ) ( )P P P P P≡ ++ + − − − +− − − +where
( , )P a b a b= − i 0.71 1 0.71≤ − violation!QM: For 0°, 90°, and 45°:
Experiment (Aspect et al., 1982; photons instead of spins, CHSH):yes, “spooky action-at-a-distance”
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University of California, RiversideAlexander Korotkov
What about causality?Actually, not too bad: you cannot transmit your own information
choosing a particular measurement direction aa Result of the other
measurement does notdepend on direction a
Randomness saves causalityor
Collapse is still instantaneous: OK, just our recipe, not an “objective reality”, not a “physical” process
Consequence of causality: No-cloning theorem
You cannot copy an unknown quantum stateOtherwise get information on direction a (and causality violated)
Wootters-Zurek, 1982; Dieks, 1982; Yurke
Proof:
Application: quantum cryptographyInformation is an important concept in quantum mechanics
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University of California, RiversideAlexander Korotkov
Quantum measurement in solid-state systems
No violation of locality – too small distances
However, interesting informational aspects of continuous measurement (gradual collapse)
Starting point: qubit
detectorI(t)
What happens to a solid-state qubit (two-level system)during its continuous measurement by a detector?
How qubit evolution is related to the noisy detectoroutput I(t)?
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University of California, RiversideAlexander Korotkov
Superconducting “charge” qubits
Single Cooperpair box
Quantum coherent (Rabi) oscillations
2e
Vg
n+1
nEJ
22(2 )ˆ ( )
2(| 1 | | 1 |)
2
ˆJ
geH nCE n n n n
n⟩ ⟨ + + + ⟩ ⟨
= -
-
Nakamura, Pashkin, Tsai (Nature, 1998)
2 gn
∆t (ps)
Vion et al. (Devoret’s group); Science, 2002Q-factor of coherent (Rabi) oscillations = 25,000
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University of California, RiversideAlexander Korotkov
Superconducting “charge” qubits (2)Duty, Gunnarsson, Bladh,
Delsing, PRB 2004Guillaume et al. (Echternach’s
group), PRB 2004
2e
Vg V I(t)
Cooper-pair boxmeasured by single-electron transistor (SET)(actually, RF-SET)
Setup can be used for continuous measurements
All results are averaged over many measurements (not “single-shot”)
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University of California, RiversideAlexander Korotkov
Semiconductor (double-dot) qubitT. Hayashi et al., PRL 2003
Detector is not separated similar to Nakamura-98, also possible to use a separate detector
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University of California, RiversideAlexander Korotkov
“Which-path detector” experiment
2 2( )(1 )
( )4 I
eV Th T T
IS
∆Γ = = ∆−
∆I – detector response, SI – shot noise
Buks, Schuster, Heiblum, Mahalu, and Umansky, Nature 1998
Dephasing rate:
The larger noise, the smaller dephasing!!!
(∆I)2/4SI ~ rate of “information flow”
τm= 2SI /(∆I)2 – “measurement time”Theory: Aleiner, Wingreen,and Meir, PRL 1997 (Shnirman-Schon, 1998)
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University of California, RiversideAlexander Korotkov
The system we consider: qubit + detector
eH
I(t)Double-quantum-qot (DQD) and
quantum point contact (QPC)
qubit
detectorI(t)
2e
Vg V
I(t)
Cooper-pair box (CPB) andsingle-electron transistor (SET)
H = HQB + HDET + HINT
ε – asymmetry, H – tunnelingHQB = (ε/2)(c1+c1– c2
+c2) + H(c1+c2+c2
+c1)
Ω = (4H 2+ε2)1/2/Ñ – frequency of quantum coherent (Rabi) oscillations
Two levels of average detector current: I1 for qubit state |1⟩, I2 for |2⟩Response: ∆I= I1–I2 Detector noise: white, spectral density SI
† † † †, ( )DET r r r r rl l l l ll r l rH E a a E a a T a a a a= + ++∑ ∑ ∑
† † † †1 1 2 2, ( ) ( )INT r rl ll rH T c c c c a a a a= ∆ − +∑ 2IS eI=
DQD and QPC(setup due to Gurvitz, 1997)
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University of California, RiversideAlexander Korotkov
1 01 10 02 2
1 1 0 02 2 0 1
What happens to a qubit state during measurement?e
H
I(t)For simplicity (for a moment) H=ε=0 (infinite barrier),evolution due to measurement only
“Orthodox” answer1 1 1 exp( ) 1 02 2 2 2 21 1 exp( ) 1 102 2 2 2 2
t
t
−Γ
−Γ
→ →
|1> or |2>, depending on the result
“Conventional” (decoherence) answer (Leggett, Zurek)
no measurement result! ensemble averaged
Orthodox and decoherence answers contradict each other!
yesyesBayesian, 1998
yesnoConventional (ensemble)
noyesOrthodoxContinuous measurementsSingle quantum systemsapplicable for:
Bayesian formalism describes gradual collapse of a single quantum systemNoisy detector output I(t) should be taken into account
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University of California, RiversideAlexander Korotkov
Bayesian formalism for a single qubite
H
I(t) |1Ò Æ I1, |2Ò Æ I2 ∆I=I1-I2 , I0=(I1+I2)/2, SI – detector noise
† † † †1 1 2 2 1 2 2 1
ˆ ( ) ( )2QBH c c c c H c c c cε= − + +
(A.K., 1998)
12 11 22 011 22
12 11 22 12 11 22 0 1212
2( / ) Im (2 / )[ ]
( / ) ( / ) ( ) ( ) ( / )[ ]
( )
( )
I
I
H I S I
i i H I S I
I t
I t
ρ ρ ρ ρ ρ
ρ ε ρ ρ ρ ρ ρ ρ γρ
• •
•
∆
+ ∆
= - = - + -
= + - - - -
2
2
( ) / 4 ,
1 / ( ) / 4I
I
I S
I S
γη γ η
Γ ∆ Γ −
Γ ∆ Γ − ≤
ensemble decoherencedetector ideality (efficiency), 100%
= -
= - =
Ideal detector (η=1) does not decohere a single qubit; then random evolution of qubit wavefunction can be monitored
For simulations: 0 22 11( ) = ( ) / 2 ( ), = II t I I t S Sξρ ρ ξ∆- - +
Averaging over ξ(t) ï conventional master equation
Similar formalisms developed earlier. Key words: Imprecise, weak, selective, or conditional measurements, POVM, Quantum trajectories, Quantum jumps, Restricted path integral, etc.
Names: Davies, Kraus, Holevo, Mensky, Caves, Gardiner, Carmichael, Plenio, Knight, Walls, Gisin, Percival, Milburn, Wiseman, Onofrio, Habib, Doherty, etc. (incomplete list)
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University of California, RiversideAlexander Korotkov
Assumptions needed for the Bayesian formalism
• Detector voltage is much larger than the qubit energies involvedeV >> ÑΩ, eV >> ÑΓ (no coherence in the detector,
Ñ/eV << (1/Ω, 1/Γ); Markovian approximation)
• Small detector response, |∆I | << I0, ∆I=I1- I2, I0 = (I1+ I2)/2Many electrons pass through detector before qubit evolves noticeably.(Not a really important condition, but simplifies formalism.)
Coupling C ~ Γ/Ω is arbitrary [we define C =Ñ(∆I )2/SIH ]
11 22 12 11 22 0
12 12 11 22 12 11 22 0 12
2= = 2 Im [ ( ) ]
= ( ) ( ) [ ( ) ]
I
I
d d H I I t Idt dt Sd H Ii i I t Idt S
ρ ρ ρ ρ ρ
ερ ρ ρ ρ ρ ρ ρ γρ
∆
∆
- - + -
+ - + - - -
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University of California, RiversideAlexander Korotkov
“Quantum Bayes theorem“ (ideal detector assumed)
eH
I(t)
|1> |2>
2
2
11 1 22 2
1 2
01 ( )
( , ) (0) ( , ) (0) ( , )1( , ) exp[ ( ) / 2 ],
2/ 2 , | | , /II i i
i i
I I t dt
P I P I P I
P I I I DD
D S I I I S I
τττ ρ τ ρ τ
τπ
τ τ
≡
= +
= − −
= −
∫
11 12
21 22
(0) (0)(0) (0)
ρ ρρ ρ
H = ε = 0(“frozen” qubit)
( ) ( | )( | )
( ) ( | )k kk
i ii
P B P A BP B A
P B P A B=∑
After the measurement during time τ, the probabilities should be updated using the standard Bayes formula:
Measurement (during time τ):
I_
P
I1 I2
Iactual_
2D1/2 2D1/2
Initial state:
211 1
11 2 211 1 22 2
12 1222 111/2 1/2
12 22 12 22
(0) exp[ ( ) / 2 ]( )(0) exp[ ( ) / 2 ] (0) exp[ ( ) / 2 ]
( ) (0) , ( ) 1 ( )[ ( ) ( )] [ (0) (0)]
I I DI I D I I Dρρ τ
ρ ρρ τ ρ ρ τ ρ τ
ρ τ ρ τ ρ ρ
+- -
=- - - -
= = -
Quantum Bayesformulas:
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University of California, RiversideAlexander Korotkov
“Informational” derivation of the Bayesian formalism
Step 1. Assume H = ε = 0, “frozen” qubit Since ρ12 is not involved, evolution of ρ11 and ρ22 should be the same as in the classical case, i.e. Bayes formula (correspondence principle).
Step 2. Assume H = ε = 0 and pure initial state, ρ12 (0) = [ρ11(0) ρ22(0)]1/2
For any realization |ρ12 (t)| ≤ [ρ11(t) ρ22(t)]1/2 . Hence, averaging over ensemble of realizations gives |ρ12
av(t)| ≤ ρ12av(0) exp[æ (∆I 2/4SI) t]
However, conventional (ensemble) result (Gurvitz-1997, Aleiner et al.-1997) for QPC is exactly the upper bound: ρ12
av (t) = ρ12av (0) exp[æ (∆I2/4SI) t].
Therefore, pure state remains pure: ρ12 (t) = [ρ11(t) ρ22(t)]1/2.
Step 3. Account of a mixed initial stateResult: the degree of purity ρ12 (t) / [ρ11(t) ρ22(t)]1/2 is conserved.
Step 4. Add qubit evolution due to H and ε.
Step 5. Add extra dephasing due to detector nonideality (i.e., for SET).
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University of California, RiversideAlexander Korotkov
“Microscopic” derivation of the Bayesian formalism
qubit detector pointerquantum interaction
frequentcollapse
classicalinformation
( )nij tρ ( )kn t
n – number of electronspassed through detector
Detector collapse at t = tkParticular nk is chosen at tk
Schrödinger evolution of “qubit + detector”for a low-T QPC as a detector (Gurvitz, 1997)
11 22( ) ( ) ( )kn n
kP n t tρ ρ= +11 111 11 11 12
12 222 22 22 12
11 21 212 12 11 22 12 12
2 Im
2 Im
( )2
n n n n
n n n n
n n n n n n
I Id Hdt e e
I Id Hdt e e
I II Id Hi idt e e
ρ ρ ρ ρ
ρ ρ ρ ρ
ερ ρ ρ ρ ρ ρ
−
−
−
= − + −
= − + +
+= + − − +
,
11 22
( 0) = ( 0)
( )( 0) =
( ) ( )
nij k n nk ij k
nkij k
ij k nk nkk k
t t
tt
t t
ρ δ ρ
ρρ
ρ ρ
+ +
++
11 1 22 211 22
11 1 22 2 11 1 22 21/ 2
11 2212 12 1/ 2
11 22
(0) ( ) (0) ( )( ) , ( )(0) ( ) (0) ( ) (0) ( ) (0) ( )
( / )[ ( ) ( )]( ) (0) , ( ) exp( / ),![ (0) (0)]
ni
i i
P n P nt tP n P n P n P n
I t et tt P n I t en
ρ ρρ ρρ ρ ρ ρ
ρ ρρ ρρ ρ
= =+ +
= = −
If = = 0,H εthis leads to
which are exactly quantum Bayes formulas
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University of California, RiversideAlexander Korotkov
One more derivation
Translating well-developed “quantum trajectory” formalism from quantum optics into solid-state language
(equivalent though looks very different)
Goan and Milburn, 2001Also: Wiseman, Sun, Oxtoby, Warszawsky,
Polkinghorne, etc.
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University of California, RiversideAlexander Korotkov
Nonideal detectors with input-output noise correlation
qubit(ε, H)
idealdetector
signal
quantumbackaction
noise
+I(t)S0+S1
ξ2(t) = Aξ1(t)
ξ3(t)
fully correlated ξ1(t)
S1
Id (t)S0 classical
current
classical noiseaffecting ε
classical noiseaffecting ε
detector
11 22 12 11 22 0
12 12 11 22 12 11 22 0 0 12
22 Im [ ( ) ]
( ) ( ) [ ( ) ] [ ( ) ]
I
I
d d IH I t Idt dt Sd Ii iH I t I iK I t Idt S
ρ ρ ρ ρ ρ
ρ ερ ρ ρ ρ ρ ρ γρ
∆= − = − + −
∆= + − + − − + − −
A.K., 2002
1 00 1, I
I
AS SK S S S
Sθ+
= = +
K – correlation between outputand ε–backaction noises
Fundamental limits for ensemble decoherenceΓ = γ + (∆I)2/4SI , γ ≥ 0 ⇒ Γ ≥ (∆I)2/4SI
Γ = γ + (∆I)2/4SI + K2SI/4 , γ ≥ 0 ⇒ Γ ≥ (∆I)2/4SI + K2SI /4
Translated into energy sensitivity: (ЄI ЄBA)1/2 ≥ /2 or (ЄIЄBA – ЄI,BA2)1/2 ≥ /2
(known since 1980s)
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University of California, RiversideAlexander Korotkov
Ideality of solid-state detectors(ideal detector does not cause single qubit decoherence)
1. Quantum point contact Theoretically, ideal quantum detector, η=1A.K., 1998 (Gurvitz, 1997; Aleiner et al., 1997)
I(t)Experimentally, η > 80%
(using Buks et al., 1998)
Very non-ideal in usual operation regime, η ‹‹12. SET-transistorShnirman-Schőn, 1998; A.K., 2000, Devoret-Schoelkopf, 2000
However, reaches ideality, η = 1 if:- in deep cotunneling regime (Averin, vanden Brink, 2000)- S-SET, using supercurrent (Zorin, 1996)- S-SET, double-JQP peak (η ~ 1) (Clerk et al., 2002)- resonant-tunneling SET, low bias (Averin, 2000)
I(t)
Can reach ideality, η = 13. SQUID 4. FET ?? HEMT ??ballistic FET/HEMT ??
V(t)
(Danilov-Likharev-Zorin, 1983;Averin, 2000)
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University of California, RiversideAlexander Korotkov
Bayesian formalism for N entangled qubits measured by one detector
qb 1
detector
qb 2 qb … qb N
I(t)
ρ (t)Up to 2N levels of current
( ( ) )( )2
]k jj k ij ij
I II t I I γ ρ
++ − − −
1ˆ[ , ] ( ( ) )( )2
[k
k iij qb ij ij kk i k
I Id i H I t I Idt S
ρ ρ ρ ρ +−= + − − +∑
1 2( 1)( ) / 4 ( ) ( ) ( )i
Iij i j ii iI I S I t t I tγ η ρ ξ−= − − = +∑Averaging over ξ(t) î master equation
(Stratonovich form)
No measurement-induced dephasing between states |iÒ and |jÒ if Ii = Ij !A.K., PRA 65 (2002),
PRB 67 (2003)
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University of California, RiversideAlexander Korotkov
Measurement vs. decoherenceWidely accepted point of view:
measurement = decoherence (environment)
Is it true?• Yes, if not interested in information from detector
(ensemble-averaged evolution)
• No, if take into account measurement result(single quantum system)
Measurement result obviously gives us more information about the measured system, so we know its quantum state better (ideally, a pure state instead of a mixed state)
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University of California, RiversideAlexander Korotkov
Some experimental predictions and proposals
• Direct experimental verification (1998)
• Measured spectral density of Rabi oscillations (1999, 2000, 2002)
• Bell-type correlation experiment (2000)
• Quantum feedback control of a qubit (2001)
• Entanglement by measurement (2002)
• Measurement by a quadratic detector (2003)
• Simple quantum feedback of a qubit (2004)
• Squeezing of a nanomechanical resonator (2004)
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University of California, RiversideAlexander Korotkov
Direct verification of the Bayesian evolution(A.K., 1998)
Idea: check the predicted evolution of an almost pure qubit state
0.0
0.5
1.0ρ11
t
Reρ12
0 5 10 15 20 25 30
-0.5
0.0
0.5
1.0
ρ11Re ρ12Im ρ12
Evolution from 1/2-alive to Density matrix purification1/3-alive Schrödinger cat by measurement
time
stop & check
1. Start with completely mixed state.2. Measure and monitor the Rabi phase.3. Stop evolution (make H=0) at state |1›. 4. Measure and check.
1. Prepare coherent state and make H=0.2. Measure for a finite time t.3. Check the predicted wavefunction (using
evolution with H≠0 to get the state |1›.
Difficulty: need to record noisy detector current I(t) and solve Bayesianequations in real time; typical required bandwidth: 1-10 GHz.
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University of California, RiversideAlexander Korotkov
Measured spectrum of qubit coherent oscillations
qubit detectorI(t)
α What is the spectral density SI (ω)of detector current?
A.K., LT’99Averin-A.K., 2000A.K., 2000Averin, 2000Goan-Milburn, 2001Makhlin et al., 2001Balatsky-Martin, 2001Ruskov-A.K., 2002 Mozyrsky et al., 2002 Balatsky et al., 2002Bulaevskii et al., 2002Shnirman et al., 2002Bulaevskii-Ortiz, 2003Shnirman et al., 2003
2 2
0 2 2 2 2 2( )( )
( )IIS Sω
ω ωΩ ∆ Γ= +− Ω + Γ
Spectral peak can be seen, butpeak-to-pedestal ratio ≤ 4η ≤ 4
1 200, ( ) / 4I Sε η −= Γ = ∆2( ) / IC I HS= ∆
0.0 0.5 1.0 1.5 2.00123456
ω /Ω
S I(ω)/S
0
α = 0.1η = 1
ε /H=0 1 2 classical
limit
0.0 0.5 1.0 1.5 2.00
2
4
6
8
10
12
ω/Ω
I(ω)/S
0
C=13
10
31
0.3
Assume classical output, eV » Ω
S
(result can be obtained using variousmethods, not only Bayesian method)
Weak coupling, α = C/8 « 1
02 2
0 2 2 2 2/( )
1 ( / 4 )IS HS S
Hη εω
ω= +
+ Ω Γ2 2 1
02 2 2 2
4 (1 / 2 )1 [( ) (1 2 / )]
S HH
η εω
−++
+ −Ω Γ − Ω
Contrary:Stace-Barrett, 2003
(PRL 2004)
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University of California, RiversideAlexander Korotkov
Possible experimental confirmation?Durkan and Welland, 2001 (STM-ESR experiment similar to Manassen-1989)
p e a k 3 . 5n o i s e
≤
(Colm Durkan,private comm.)
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University of California, RiversideAlexander Korotkov
Somewhat similar experiment“Continuous monitoring of Rabi oscillations in a Josephson flux qubit”
E. Il’ichev et al., PRL, 20031 ( ) cos2 HFx z zH W tσ ε σ σ ω= ∆ +- -
2 2 ; 0)( HFω ε ε≈ ∆ + ≠
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University of California, RiversideAlexander Korotkov
Bell-type measurement correlation on
offτA τB
τ onoff
QA= ∫IAdt QB= ∫IBdt
(A.K., 2000)
0 1 2 3-0.5
-0.25
0
0.25
0.5
τ Ω /2π
δ B =
(<Q
B> -
Q0B
)/∆Q
B
τA(∆ IA)2/SA = 1
(QA/τA- I 0A) /∆ IA=
δQA> 0 δQA= 0 δQA> 0
QA is fixed (selected)0.6, 0, -0.3
conventional 0 1 2 3
-0.5
-0.25
0
0.25
0.5
τ Ω /2π
after π/2 pulsedetector A qubit detector B
Idea: two consecutive finite-time (imprecise) measurements of a qubit by two detectors; probability distribution P(QA, QB, τ) shows the effect of the first measurement on the qubit state.
Proves that qubit remains in a pure state during measurement (for η=1)
Advantage: no need to record noisy detector output with GHz bandwidth;instead, we use two detectors and fast ON/OFF switching.
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University of California, RiversideAlexander Korotkov
Quantum feedback control of a qubitSince qubit state can be monitored, the feedback is possible!
qubit
H
e
detector Bayesian equations
I(t)
control stage
(barrier height)
ρij(t)
comparison circuit
desired evolution
feedback
signal
environment
C<<1
Goal: maintain desired phase of coherent (Rabi) oscillationsin spite of environmental dephasing (keep qubit “fresh”)
Ruskov & A.K., 2001
Hqb= HσX
Idea: monitor the Rabi phase φ by continuous measurement and apply feedback control of the qubit barrier height, ∆HFB/H = −F×∆φ
To monitor phase φ we plug detector output I(t) into Bayesian equations
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University of California, RiversideAlexander Korotkov
Performance of quantum feedback(no extra environment)
Qubit correlation function Fidelity (synchronization degree)
2
-1a
desir
( ) / couplingavailable bandwidth
F feedback strength D= 2 Tr 1
IC I S Hτ
ρρ
= ∆ −−
−⟨ ⟩ −
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.0
0.2
0.4
0.6
0.8
1.0
τa / T=2/3
1/30
"direct"
F (feedback factor)
D (
sync
hron
izat
ion
degr
ee)
C=1η =1de=0
For ideal detector and wide bandwidth,fidelity can be arbitrarily close to 100%
D = exp(−C/32F)
2 /cos( ) exp ( 1)2 16
FHz
t CK eF
ττ −Ω = − (for weak coupling and good fidelity)
Detector current correlation function2
2 /
2 /
( ) cos( ) (1 )4 2
exp ( 1) ( )16 2
FHI
FH I
I tK e
SC eF
τ
τ
τ
δ τ
−
−
∆ Ω= +
× − +
0 5 10 15-0.50
-0.25
0.00
0.25
0.50
τ Ω /2π
Kz(
τ )
C=1, η=1, F=0, 0.05, 0.5
Ruskov & Korotkov, PRB 66, 041401(R) (2002)
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University of California, RiversideAlexander Korotkov
Suppression of environment-induced decoherence by quantum feedback
0 1 2 3 4 5 6 7 8 9 100.80
0.85
0.90
0.95
1.00
F (feedback factor)
D (
sync
hron
izat
ion
degr
ee)
Cenv /Cdet= 0 0.1 0.5
C=Cdet=1τa=0
envmax
det1
2CDC
≈ −
0.00 0.05 0.100.95
1.00
0 1 2 3 4 5 6 7 8 9 100.0
0.2
0.4
0.6
0.8
1.0
Dm
ax (s
ynch
roni
zatio
n)
Cenv/Cdet (relative coupling to environment)
Example: if qubit coupling to environment is 10 times weaker than to detector,then Dmax = 95% and qubit fidelity 97.5%. (D = 0 without feedback.)
Experimental problems:• necessity of very fast real-time solution
of the Bayesian equations • wide bandwidth (>>Ω, GHz-range) of the line
delivering noisy signal I(t) to the “processor”
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University of California, RiversideAlexander Korotkov
Simple quantum feedback of a solid-state qubit(A.K., cond-mat/0404696)
detectorI(t)
×cos(Ω t), τ-average
phas
e
X
Y
φmqubit
H =H0 [1– F×φm(t)]control
×sin(Ω t), τ-average
Hqb= HσX
C <<1local oscillator
Goal: maintain coherent (Rabi) oscillations forarbitrary long time
Idea: use two quadrature components of the detector current I(t)to monitor approximately the phase of qubit oscillations(a very natural way for usual classical feedback!)
0( ) [ ( ') ] cos( ') exp[ ( ') / ]t
X t I t I t t t dtτ−∞
= − Ω − −∫0( ) [ ( ') ] sin( ') exp[ ( ') / ]
tY t I t I t t t dtτ
−∞= − Ω − −∫
arctan( / )m Y Xφ = −
(similar formulas for a tank circuit instead of mixing with local oscillator)
Advantage: simplicity and relatively narrow bandwidth (1 / ~ )dτ Γ << ΩAnticipated problem: without feedback the spectral peak-to-pedestal ratio <4,
therefore not much information in quadratures(surprisingly, situation is much better than anticipated!)
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University of California, RiversideAlexander Korotkov
Accuracy of phase monitoring via quadratures(no feedback yet)
0 1 2 3 4 5 6 7 80.0
0.5
1.0
1.5
2.0
C = 0.1, 0.3, 13
1030
∆φrm
s
τ [(∆I)2/SI]
π/31/2
no noise
C<<1
(averaging time)
(pha
se in
accu
racy
)
uncorrelated noise C<<1
C – dimensionless coupling 1/Γd=4SI/(∆I)2
-3 -2 -1 0 1 2 30
1
2
41
8
τ [(∆I)2/SI] =
∆φ
p (∆φ
)
C = 0.1
2.16
(non-Gaussiandistributions)
∆φ=φ-φm
weak coupling C<<1
Noise improves the monitoring accuracy!(purely quantum effect, “reality follows observations”)
0/ [ ( ) ]sin( ) ( / )Id dt I t I t I Sφ φ= − − Ω + ∆2 2 1/ 2
0/ [ ( ) ]sin( ) /( )m md dt I t I t X Yφ φ= − − Ω + +(actual phase shift, ideal detector)
(observed phase shift)
Noise enters the actual and observed phase evolution in a similar way
Quite accurate monitoring! cos(0.44)≈0.9
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University of California, RiversideAlexander Korotkov
Simple quantum feedback
0.0 0.1 0.2 0.3 0.40.0
0.2
0.4
0.6
0.8
1.0
12
F/C
D, <
X>(
4/τ ∆
I)
48
0.50.2
τ[(∆I)2/SI]=
0.1C = 0.1η = 1
classical feedback
(feedback strength)
(fide
lity)
fidelity for different averaging τ
weak coupling C
D – feedback efficiency2 1
Tr ( ) ( )Q
Q des
D F
F t tρ ρ≡ −
≡ ⟨ ⟩
Dmax ≈ 90%
(FQ ≈ 95%)
How to verify feedback operation experimentally?Simple: just check that in-phase quadrature ⟨X⟩
of the detector current is positive (4 / )D X Iτ= ⟨ ⟩ ∆
⟨X⟩=0 for any non-feedback Hamiltonian control of the qubit
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University of California, RiversideAlexander Korotkov
Effect of nonidealities- nonideal detectors (finite
quantum efficiency η)and environment
- qubit energy asymmetry ε- frequency mismatch ∆Ω
• Fidelity FQ up to ~95% achievable (D~90%)• Natural, practically classical feedback setup• Averaging τ~1/Γ>>1/Ω (narrow bandwidth!)• Detector efficiency (ideality) η ~0.1 still OK• Robust to asymmetry ε and frequency shift ∆Ω• Simple verification: positive in-phase quadrature ⟨X⟩
Simple enoughexperiment?!
0.0 0.2 0.4 0.6 0.80.0
0.2
0.4
0.6
0.8
1.0ηeff =
0.5
0.2
0.1
ε/H0= 10.5
0
∆Ω/CΩ=0.2
0
C = 0.1τ [(∆I)2/SI] = 1
1
0.1
F/C (feedback strength)D
(feed
back
effi
cien
cy)
Quantum feedbackstill works quite well
Main features:
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University of California, RiversideAlexander Korotkov
Quantum feedback in opticsRecent experiment: Science 304, 270 (2004)
First detailed theory:H.M. Wiseman and G. J. Milburn, Phys. Rev. Lett. 70, 548 (1993)
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University of California, RiversideAlexander Korotkov
Two-qubit entanglement by measurement
Ha Hb
DQDa QPC DQDb
I(t)
Ha Hb
Vga VgbV
qubit a qubit bSET
I(t)qubit 1 qubit 2
detectorI(t)
entangled
ρ (t)
Collapse into |BellÚ state (spontaneous entanglement) with probability 1/4 starting from fully mixed state
Ruskov & A.K., 2002
0 10 20 30 40 50 60 700.0
0.2
0.4
0.6
0.8
1.0
entangled, P=1/4
oscillatory, P=3/4
Ω t
ρ Bel
l (t)
C = 1η = 1
0 1 202468
1012
ω /Ω
S I(ω)/S
0
0 1 2024
ω /ΩS I(ω)/S
0
Two evolution scenarios:
Symmetric setup, no qubit interaction
Peak/noise= (32/3)η
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University of California, RiversideAlexander Korotkov
Quadratic quantum detectionMao, Averin, Ruskov, Korotkov, PRL-2004
Ha Hb
Vga VgbV
qubit a qubit bSET
I(t)
Peak only at 2Ω, peak/noise = 4η2 2
0 2 2 2 2 24 ( )( )
( 4 )IIS Sω
ω ωΩ ∆ Γ= +
− Ω + Γ
Ibias
V(f)
ω/Ω0 1 2 30246
S I(ω
)/S0
0 1 2 30246
ω/Ω
S I(ω
)/S0
quadraticI, V
q0,φ
Nonlinear detector:spectral peaks at Ω, 2Ω and 0
Quadratic detector:
Three evolution scenarios: 1) collapse into |↑↓ -↓↑ Ú, current IÆ∞, flat spectrum2) collapse into |↑↑ - ↓↓ Ú, current IÆÆ, flat spectrum; 3) collapse into remaining subspace, current (IÆ∞+ IÆÆ)/2, spectral peak at 2Ω
Entangled states distinguished by average detector current
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University of California, RiversideAlexander Korotkov
Some experiments on nanoresonatorsLaHaye, Buu, Camarota, and Schwab, Science-2004
f = 20 MHz
∆x = 5.8∆x0 3.8 fm/Hz1/2
Knobel, Cleland, Nature-2003
f = 117 MHz
2 fm/Hz1/2
∆x ~ 100 ∆x0
Ming et al. (Roukes’ group), Nature-2003
f = 1.03 GHz
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University of California, RiversideAlexander Korotkov
QND squeezing of a nanomechanical resonatorRuskov, Schwab, Korotkov, cond-mat/0406416,
I(t)
m,ω0
∼V(t)
x
QPC
resonator ,
† † †ˆ ( . .)DETrl l r
r r r rl l l lH E a a E a a Ma a H c= + + +∑ ∑ ∑†
,
ˆ ˆ( . .)INT rll r
H M x a a H c= ∆ +∑
2 2 20 0
ˆ ˆ ˆ/ 2 / 2H p m m xω= + cond-mat/0411617
Experimental status:
Continuous monitoring and quantum feedback can cool nanoresonatordown to the ground state (Hopkins, Jacobs, Habib, Schwab, PRB 2003)
ω0/2π∼ 1 GHz ( ω0 ∼ 80 mK), Roukes’ group, 2003
∆x/∆x0 ∼ 5 [SQL ∆x0=( /2mω0)1/2], Schwab’s group,2004
Our paper: Braginsky’s stroboscopic QND measurement usingmodulation of detector voltage ⇒ squeezing becomes possible
Potential application: ultrasensitive force measurements
Other most important papers:Doherty, Jacobs, PRA 1999 (formalism for Gaussian states) Mozyrsky, Martin, PRL 2002 (ensemble-averaged evolution)
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University of California, RiversideAlexander Korotkov
Stroboscopic QND measurementsQuantum nondemolition (QND) measurements (Braginsky-Khalili book)(a way to suppress measurement backaction and overcome standard quantum limit)Idea: to avoid measuring the magnitude conjugated to the magnitude of interest
Standard quantum limit/ 2p x∆ > ∆
1( )x t 2( )x t
Example: measurement of x(t2)-x(t1)First measurement: ∆p(t1)> /2∆x(t1), then even for accurate second measurement
inaccuracy of position difference is ∆x(t1)+ (t2-t1) /2m∆x(t1)> (t2-t1) /21/2m
Stroboscopic QND measurements (Braginsky et al., 1978; Thorne et al., 1978)
oscillatorIdea: second measurement exactly one oscillation
period later is insensitive to ∆p(or ∆t =nT/2, T=2π/ω0)
• continuous measurement• weak coupling with detector• quantum feedback to suppress “heating”
Difference in our case:
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University of California, RiversideAlexander Korotkov
Squeezing by stroboscopic (pulse) modulation
0
102 / n=
141 142 143 144 145
1
2
3
4
5
6
7
8
0 / 2tω π
xD
xD⟨ ⟩
<< xxD D⟨ ⟩
0/ω ω0.0 0.5 1.0 1.5 2.0 2.5
0
2
4
6
8
10
12
(∆x 0
/ ∆x)
2
212/31/22/5 C = 0.5δt /T0= 0.05 η = 1
141 142 143 144 145
0
2
4
6
8pp xxD D D
0 / 2tω π
Dx,
D⟨x⟩
Sque
ezin
g S
usingfeedback
Dx=(∆x)2
0 200 400 6000
5
10
15
20
25
N
S N
C0 = 1, δt /T0= 0.025
C0 = 10.5
0.1
δt /T0= 0.05
η = 1ω =2ω0Amod = 1
inst. meas.(δt=0)
pulse modulation ω ω Squeezing buildup (in time)
f(t)
Efficient squeezing at ω=2ω0/nSá1
(natural QND condition)Ruskov-Schwab-Korotkov
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University of California, RiversideAlexander Korotkov
Squeezing by stroboscopic modulationAnalytics (weak coupling, short pulses)
0
1
f(t)
Maximum squeezing Linewidth
1.95 2.00 2.0502468
1012
0.1
C = 10.5
Sque
ezin
g S 00.05t Tδ =
1η =
0/ω ω
03 4
02
4 ( )3
C tnδ ω
ωπ η
∆ =00
2 3(2 / )S ntηω
ω δ=
C0 – dimensionless coupling with detectorδt – pulse duration, T0= 2π/ω0η – quantum efficiency of detector
20 0~ 3 / ( )C tη ω δSqueezing requires pulses
Sque
ezin
g S m
ax
1
10
100
T/ hω0 =
10100
10 100 1000 10 10 10 4 5 6
1
η = 1ω = 2 ω0Amod = 1
0
0C Qη
Finite Q-factor and finite temperature limitmaximum squeezing Smax
max0
0
1/334 coth( / 2 )
C QS
Tη
ω
=
(So far in experiment η1/2C0Q~0.1)
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University of California, RiversideAlexander Korotkov
Conclusions
Continuous quantum measurement is not equivalent to decoherence (environment) if detector output (information)is taken into account, in contrast to ensemble-averaged case
Bayesian approach to continuous quantum measurementis a simple, but new and interesting subject in solid-statemesoscopics
Several experimental predictions have been already made;however, many problems not studied yet
No direct experiments yet (few indirect ones); hopefully,coming soon