Mesoscopic quantum measurements

Transcription

1 Mesoscopic quantum measurements D.V. Averin Department of Physics and Astronomy SUNY Stony Brook A. Di Lorentzo K. Rabenstein V.K. Semenov D. Shepelyanskii E.V. Sukhorukov

2 Summary α β Example of the trivial wavefunction reduction: Basic understanding of quantum measurements: the measurement process is ``decoherence with an output ; reduction of the wave-function is a classically trivial act of picking one random outcome out of several possibilities. ``Quantum measurement problem : wave function is typically reduced in an unusual way not compatible with the Schrödinger equation.

3 Outline. Basics - superconductor and semiconductor single-charge qubits - QPC as a quantum detector - ideal quantum detectors: trade-off between information gain and back-action dephasing in quantum measurements. Dynamics of quantum measurements - linear - quadratic

4 R.A. Millikan Phys. Rev. (Ser. ) (9).

5 Quantum dynamics of Josephson junctions Superconductor can be thought of as a BEC of Cooper pairs: one single-particle state Ψ = iϕ ne occupied with macroscopic number of particles. The phase φ and the number of particles n are conjugate quantum variables (Anderson 64; vanchenko Zil berman 65): [nϕ] =i. This relation describes dynamics of addition or removal of particles to/from the condensate.

6 f quantum fluctuations of phase φ become large junction behavior can be described as a semiclassical dynamics of charge that leads to controlled transfer of individual Cooper pairs (D.V. Averin A.B. Zorin and K.K. Likharev 985): H = E / ϕ E cosϕ = = E C C ( n q) E J J ( n n ± n ± n ) E C (e) C.. (a) E C /E J = (b) ε /E J 6 3 <n> q=v g C/e q

7 Cooper-pair box as charge qubit f a small Josephson junction is biased through a capacitor not resistor one obtains configuration of a ``Cooper-pair box (M. Buttiker 987 H. Pothier et al. 989) which avoids the problem of electromagnetic environment. For E J <<E C and q / the charge tunneling dynamics in the junction is directly reduced to the two-state form: H = E C ( q ) σ ( E ) σ z J x

8 a (pa) Coherent oscillations in two coupled charge qubits a reservoir reservoir probe coupling island probe (pa) GHz box box dc gate pulse gate dc gate µ m b V b E J CJ E C C b C g V g C p C m V p C p V g J J C b Cg V b b c (pa) d (pa) t(ns) Yu. A. Pashkin et al. Nature 4 83 (3). 9. GHz 3 f(ghz)

9 Oscillations of probability p j to have a Cooper on the box j=: E 4 J EJ Em p () = cos( Ωt)cos( εt) sin( Ωt)sin( εt) 4Ωε Ω = ( ) / E m ( ) / δ = ( E E ). ε = δ E m = ( EJ EJ). J J Entanglement between the qubits in the process of oscillations E = p± log p± p± = ± r ± E.5 r = 4 Em Em Em [ sin ( Ωt) sin ( εt) sin ( Ωt)sin ( εt) Ω ε Ω ε Ω m 4 E sin 4 δ ( Ωt) ε Em 4 sin 4 ( εt) m E sin(ωt)sin(εt Ωε. )]. 5.. t /π /.

10 Quantum-dot qubits J.M. Elzerman et al. PRB (3).

11 Quantum antidot qubits G j - control gates Ω- inter-qad tunnel amplitude ε - quasiparticle "localization" energy nformation encoded in the position of a quasiparticle in the system of two antidots. Adiabatic level-crossing dynamics can be used to transfer quasiparticles between the antidots. Conditions of operation: T << Ω ε << E E quasiparticle excitation gap (Averin and Goldman )

12 Fractional Charge of Laughlin Quasiparticles V.J. Goldman and B. Su 995

13 nvariance of the quasiparticle charge V.J. Goldman et al. PRB ().

14 Antidot transport of the FQHE quasiparticles Antidot ``molecule demonstrates coherent quasiparticle transport in the double-antidot system.j.maasilta and V.J. Goldman PRL ().

15 Quantum point contacts (QPC) M. Field et al. PRL 7 3 (993). E. Buks et al. Nature (999).

16 Mesoscopic quantum measurements D.V. Averin Department of Physics and Astronomy SUNY Stony Brook A. Di Lorentzo K. Rabenstein V.K. Semenov D. Shepelyanskii E.V. Sukhorukov

17 Summary α β Example of the trivial wavefunction reduction: Basic understanding of quantum measurements: the measurement process is ``decoherence with an output ; reduction of the wave-function is a classically trivial act of picking one random outcome out of several possibilities. ``Quantum measurement problem : wave function is typically reduced in an unusual way not compatible with the Schrödinger equation.

18 Outline. Basics - superconductor and semiconductor single-charge qubits - QPC as a quantum detector - ideal quantum detectors: trade-off between information gain and back-action dephasing in quantum measurements. Dynamics of quantum measurements - linear - quadratic

19 Counting statistics in mesoscopic electron transport (a) j V (b) ε U (x) j εev Average current: Current noise: = ( ev / πh). k D k S ( ) = ( ev / πh ) Dk ( Dk ). k Full counting statistics: (one mode) P ( N ) n = C n N D n ( D) N n N = ( ev L.S. Levitov and G.B. Lesovik 993 / πh) t.

20 Counting statistics and detector properties of a QPC ( α Detector-induced ``back-action dephasing of the qubit α β ) in ( t R r L ) β ( t R r L ) ( qubit) ρ out Tr{ R L} ρout ρin ρ out ( qubit) ( r = ρ = α β α β t t r ) α β Finally summing scattering processes of different electrons we obtain back-action dephasing rate by the QPC detector at T=: Γ d = ( ev πh)ln t t r r. The dephasing rate at finite temperature: [ Γ = dε πh)ln det( f ( ε) f ( ε) S S )]. d (

21 nformation acquisition rate is determined by distinguishability of the distributions P j (n) of the transferred charge in different qubit states: W / (/ )ln [ ( ) ( )] = t P n P n n The usual counting statistics result for the QPC gives and W n / ( ) N D D R R [ P ( n) P ( n)] = )ln( ) = ev πh D D R R Γ. ( d We see that the QPC is an ideal quantum detector with W=Γ d if no information is contained in the phases of the scattering amplitudes: φ = φ φ j = arg( t j / rj ). D.V.A. and E.V. Sukhorukov 5

22 An electronic Mach-Zehnder interferometer Y. Ji et al. Nature 4 45 (3).

23 Phase information can be utilized for measurement if the QPC is included in the electronic Mach-Zender interferometer: V j r j t j χ r j iχ = i Rm Dm e Sm iχ Dm i Rm e t j The QPC can always be turned into an ideal detector by adjusting the scattering matrix S m of the second QPC of the interferometer. f all measurement information is in the phases of the scattering amplitudes: φ = φ φ D = this happens if D χ = ( φ φ) / D m = Rm = /.

24 Conditional qubit evolution Since transmission and reflection of an electron are classical alternatives evolution of the qubit density matrix can be conditioned on the particular outcome of scattering (following ``quantum-trajectories approach in quantum optics; or A.N. Korotkov 999 in solid-state context). Depending on whether electron is transmitted or reflected ] [ ] [ / * * * / D D t t D D D β α αβ αβ β α α α. ] [ ] [ / * * * / R R r r R R R β α αβ αβ β α α α or mportant points: the qubit does not decohere; probabilities to be in the states and change.

25 Bell-type relation for mesoscopic tunneling One can devise a simple sequence of qubit transformations demonstrating that charge is indeed being transferred between the qubit states even if the tunneling amplitude is vanishing. Start with the state σ x =. After transmission of an electron ψ = D D [ D D The sequence of two pulses one creating tunneling for a period of time and another changing the phase between the states : ] / ( t) dt / h = π / 4 ε( t) dt / h = tan D / D. π / returns the qubit to the initial σ x = state i.e. we have a closed cycle which involves tunneling and transfer of charge at =. For any classical charge state there is a probability p to find the state σ x =- p = min{ D D} [ D D ] /.

26 Ballistic Read-out for Flux Qubits (flux analog of the QPC detector) f = T Qubit SFQ pulses Distributed Josephson Junction Voltmeter mv H = H σ z δu ( x) U ( x) t is convenient to characterize the qubit-detector interaction by the scattering parameters t t r r.

27 Quantum dynamics of a single fluxon A. Wallraff et al. Nature (3).

28 Generic model of a mesoscopic detector Detector dynamics should exhibit macroscopically different trajectories (see e.g. J.W. Lee et al. 5) most of the actual mesoscopic detectors are based on tunneling. qubit detector Detector Hamiltonian H = ( t[ Mˆ ] ξ h. c.) Hex > > t where ξ creates excitations in the process of particle transfer between the detector reservoirs. ( n) & kl ρ ( n) * ( n) * ( n )( ) ) Γ tk tl ρkl Γ tktl ρij Γt ktl ρkl = ( Γ Γ = dt ξ( t) ξ Γ = dt ξ ( t) ξ. Ensemble-averaged evolution Conditional evolution ρ& γ kl kl = γ kl = ( Γ ρ kl Γ i[ H ρ] ) t k t j kl. ρ ( n) kl ( τ ) = ρ ()( Γ kl exp{ (/ )( Γ / Γ Γ ) n/ )( t k n (t t k l t l Γ Γ ) inϕ ) kl }.

29 Linear quantum measurements Linear-response theory enables one to develop quantitative description of the quantum measurement process with an arbitrary detector provided that it satisfies some general conditions: the detector/system coupling is weak so that the detector s response is linear; the detector is in the stationary state; the response is instantaneous. H = H S H D xf D.V.A. cond-mat/44364 cond-mat/354. S.Pilgram and M. Büttiker PRL 89 4 (). A.A. Clerk S.M. Girvin and A.D.Stone PRB (3).

30 nformation/back-action trade-off in linear measurements Dynamics of the measurement process consists of information acquisition by the detector and back-action dephasing of the measured system. The trade-off between them has the simplest form for measurements of the static system with H S =. f xj>=x j j> we have for the back-action dephasing: ρ jj' ( t) = ρ jj' () e Γ d t Γ d = π ( x x nformation acquisition by the detector is the process of distinguishing different levels of the output signal <o>=λx j in the presence of output noise S q. The signal level (and the corresponding eigenstates of x) can be distinguished on the time scale given by the by the measurement time τ m : τ m = 8πS [ λ( x q j x j' )] τ m Γ d j j' = 8( π hλ) ) S S q f S f h..

31 FDT analog for linear quantum measurements h λ 4π [ S f S q (Re S fq ) ] / where λ is the linear response coefficient of the detector S f and S q are the low-frequency spectral densities of the respectively backaction and output noise ReS fq is the classical part of their crosscorrelator. As one can see this inequality characterizes the efficiency of the trade-off between the information acquisition by the detector and back-action dephasing of the measured system. The detector that satisfies this inequality as equality is ``ideal or ``quantumlimited.

32 Continuous monitoring of the MQC oscillations The trade-off between the information acquisition by the detector and back-action dephasing manifests itself in the directly measurable quantity in the case of measurement of coherent quantum oscillations in a qubit. H = σ σ f x z H D Spectral density S o (ω) of the detector output reflects coherent quantum oscillations of the measured qubit: S o ( ω) = S q Γ d λ 4π ( ω The height of the oscillation peak in the output spectrum is limited by the link between the information and dephasing: Smax S q 4. ) Γ d ω.

33 ε=.7 S o /S q 8 Γ=. S o /S q 8 Γ/ = Ω/ = ω/..3 ω/ A.N.Korotkov and D.V.A. Phys. Rev. B ().

34 Experimental continuous monitoring of the Rabi oscillations M L L T C T S Vmax /S.5 (a) a b c ω R /ω T. (b) a b c E.l ichev et al.prl (3) (P/P ) / (P/P ) /

35 Quadratic measurements t [ Mˆ ] = t c Mˆ c ˆ M Example: DC SQUD Mˆ () z = λ σ λ σ () z t( Mˆ ) = t () () () () δ σ z δ σ z λσ z σ z Purely quadratic measurements δ j =. H =. Quadratic measurement distinguishes ``parallel from the ``antiparallel qubit state i.e. is the measurement of the product operator of the two qubits σ z () σ z (). The measurement time is τ S m a = S = ( Γ = ( / a Γ )( t λ ) / = ( Γ ) * * Γ )( t λ t λ). t( Φ) = E J ( π Φ Φ << Φ. / Φ )

36 ) 4 ( 8 ) ( ω γ ω γ ω = a S S. ) ( λ γ Γ Γ = = j j x H ) ( σ n this case quadratic measurements can entangle two non-interacting symmetric qubits. ndeed there are three possible measurement outcomes. n the first two the qubits are entangled: { } ). )( ( λ ψ ψ ψ λ λ Γ Γ = = = = = = t n the third outcome the qubits perform coherent qubit oscillations at frequency that is twice the oscillation frequency of individual qubits. W. Mao et al. PRL (4)

37 w γ Continuously-measured spectroscopy of two qubits Non-linear measurements δ j λ. H t j = ( Γ = ( = t δs z λ(sz Γ S ( ω) = S )( δ ) / λ 3 = ( ). ( ω / ) ( j) σ x ( ν ) σ z j ) () () = σ z j j γ = ( ) / 3 γ j γ. j S /S = γ ) /. S /S ν/ =.... W. Mao et al. PRB (5) ω/ (a) (b)