What is it all about?

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1 Dephasing, decoherence,... What is it all about? Consider a spinor (total phase is irrelevant) Ê Y = y eij ˆ Á Ë y Ø e -ij Average components of the spin can be expressed through the absolute values of the spinor s components and the phase Decoherence of a single spin Sz = y - y Sx = 1- Sz cosj S y = 1- Sz sinj Ø r Let us r start with, i.e. and apply magnetic field B z r S z = 0, j = 0 S x r for some time interval Dt. Spin rotation a j µ BDt B r S r ( t) Coherent Oscillations of, e.g., x-projection of the spin (!?) Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 1

2 Sz = y - y Sx = 1- Sz cosj S y = 1- Sz sinj Ø r Let us r start with, i.e. and apply magnetic field B z r S z = 0, j = 0 S x r for some time interval. Spin rotation a B r Dt j µ BDt S r ( t) Coherent Oscillations of, e.g., x-projection of the spin (!?) r One can also apply time-independent field B x r, that will cause the Zeeman splitting, and also a small oscillating in time field in z r - direction, with the frequency close to the Zeeman splitting. There will be again rotation in xy -plane a Rabi oscillations Consider a spinor (total phase is irrelevant) Ê Y = y eij ˆ Á Ë y Ø e -ij Average components of the spin can be expressed through the absolute values of the spinor s components and the phase Decoherence of a single spin Sz = y - y Sx = 1- Sz cosj S y = 1- Sz sinj Ø A random time-dependent magnetic field B r ( t) changes the phase in an uncontrollable way it rotates the spin. This can be called decoherence Relaxation times T 1, T Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03)

3 NATURE VOL APRIL 1999 NATURE VOL APRIL 1999 Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 3

4 V g gate voltage gate source drain QD current valley peak Superconducting quantum dot with D Æ e Æ e NATURE VOL APRIL 1999 peak valley Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 4

5 NATURE VOL APRIL 1999 Quantum oscillations in two coupled charge qubits Yu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin & J. S. Tsai Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 5

Quantum oscillations in two coupled charge qubits Yu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin & J. S. Tsai Quantum oscillations in two coupled charge qubits Yu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin & J. S. Tsai decoherence times 1 t coupled ª 4 t uncoupled Dr.

6 Quantum oscillations in two coupled charge qubits Yu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin & J. S. Tsai Quantum oscillations in two coupled charge qubits Yu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin & J. S. Tsai decoherence times 1 t coupled ª 4 t uncoupled Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 6

7 What is Dephasing? 1. Suppose that originally a system(an electron) was in a pure quantum state. It means that it could be described by a wave function with a given phase.. External perturbations can transfer the system to a different quantum state. Such a transition is characterized by its amplitude, which has a modulus and a phase. 3. The phase of the amplitude can be measured by comparing it with the phase of another amplitude of the same transition. Example: Fabri-Perrot interferometer beam splitter mirror 4. Usually we can not control all of the perturbations. As a result, even for fixed initial and final states, the phase of the transition amplitude has a random component. 5. We call this contribution to the phase, df, random if it changes from measurement to measurement in an uncontrollable way. 6. It usually also depends on the duration of the experiment, t: df = df(t) 7. When the time t is large enough, df exceeds p, and interference gets averaged out. 8. Definitions: df(t f ) ª p t f phase coherence time; 1/t f dephasing rate Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 7

8 Why is Dephasing rate important? Imagine that we need to measure the energy of a quantum system, which interacts with an environment and can exchange energy with it. Let the typical energy transferred between our system an the environment in time t be de(t). The total uncertainty of an ideal measurement is h environment ( t) de ææ Æ ; tæ h æææ tæ0 t! de } De ( t) ª de( t) ( *) ª fi df( t *) + t quantum uncertainty There should be an optimal measurement time t=t*, which minimizes De(t) : De(t*)= De min t* ª t! f h t ª 1 fi t * De min ª h tf de Why is Dephasing rate important? h t * ( t *) ª fi df( t *) ª 1 fi De t* ª t min f ª h t f It is dephasing rate that determines the accuracy at which the energy of the quantum state can be measured in principle. Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 8

9 How to detect phase coherence measure the dephasing/decoherence rate? Quantum phenomena in electronic systems: Weak localization Mesoscopic fluctuations Persistent current } Orthogonality catastrophe later Magnetoresistance F O No magnetic field j 1 = j O With magnetic field H j 1 - j = *p F/F 0 F = HS - magnetic flux F 0 = hc/e - through the loop flux quantum Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 9

Dephasing and Decoherence: What's It All About? Weak Localization, Magnetoresistance in Metallic Wires I 3.1 K 1.17 K DR/R 10 mk B 45 mk 10-3 f=(dt Lf<L Dtf)f 1/= L ~ 0.5 mm -0.04-0.0 0.

10 Dephasing and Decoherence: What's It All About? Weak Localization, Magnetoresistance in Metallic Wires I 3.1 K 1.17 K DR/R 10 mk B 45 mk 10-3 f=(dt Lf<L Dtf)f 1/= L ~ 0.5 mm B (T) V Mesoscopic Fluctuations; Aharonov-Bohm effect B 00mK Conductance (e /h) mk.3 90 mk mK B (Tesla) A Can we reliably extract the dephasing rate from the experiment Is energy transfer necessary for the dephasing? NO - everything that violates T-invariance Weak will destroy the constructive interference localization: EXAMPLE: random quenched magnetic field B Mesoscopic fluctuations: Therefore YES - Even strong magnetic field will not eliminate these fluctuations. It will only reduce their amplitude by factor. A Statistical analysis is unavoidable if we want to experimentally determine the dephasing rate Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 10

11 On the other hand Let the random potential change in time very slowly, but still quick on the scale of measurement time Weak localization: Mesoscopic fluctuations: B Averaging over different realizations of the disorder O At any moment j 1 = j Averaging over time A j 1 - j Is time dependent Therefore Weak localization effects do not feel the motion of impurities Mesoscopic fluctuations get suppressed Magnetic Impurities - before - after T-invariance is clearly violated, therefore we have dephasing Mesoscopic fluctuations Magnetic impurities cause dephasing only through effective interaction between the electrons. TÆ 0 Either Kondo scattering or quenching due to the RKKY exchange. In both cases no elastic dephasing Long time (spin) dynamics in metallic (spin) glasses? Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 11

A spin-echo-type technique is applied to an artificial two-level system that utilizes a charge degree of freedom in a small superconducting electrode.

12 A spin-echo-type technique is applied to an artificial two-level system that utilizes a charge degree of freedom in a small superconducting electrode. Gate-voltage pulses are used to produce the necessary pulse sequence in order to eliminate the inhomogeneity effect in the time-ensemble measurement and to obtain refocused echo signals. Comparison of the decay time of the observed echo signal with an estimated decoherence time suggests that low-frequency energy-level fluctuations due to the 1/f charge noise dominate the dephasing in the system. Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 1

13 Decoherence factor Let the spacing between the two levels, DE, fluctuate in time: ( ) ( 0) Imax t -D t I max d e ( ) ( t ) DE - DE random phase j t 1 ( t ) = Ú d ( t ) dt h 0 ( ) ( t ) e = e - D t ij Assume that d(t) is gaussian, and Ú ( 0) d ( t ) S ( w) d w DE ( wt ) 1 Ê sin ˆ D( t ) = SDE( w) Á dw h w w Ë m w m is inverse time of the measurement Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 13

14 Decoherence factor For the free induction decay For the echo signal At w Æ 0 D D FID echo 1 h ( t ) = S ( w) 1 h Ú w m DE ( ) ( 0) Imax t -D t I max ( t ) = S ( w) ( ) Ú w m DE Ê sin wt ˆ Á Æ const Ë w Even w <<t -1 are important e ( wt ) Ê sin Á Ë w ( ) ˆ ( wt 4) Ê sin Á Ë w 4 Ê sin Á Ë w 4 dw ˆ dw ( wt 4) ˆ µ w Low frequency fluctuations are not important Decoherence factor I ( ) ( 0) D FID 1 h ( t ) = S ( w) Ú w m DE ( wt ) Ê sin Á Ë w max t D( t ) e 1 Ê sin wt 4 Decho( t ) = S ( w) E Ú D Á h w 4 wm Ë max { Ê ( 0) d ( t ) S ( w) D D FID t µ t log Á Ë 1 S E ( w) µ D w t µ t I - d w E 1/f noise Decho ˆ ( ) dw ˆ dw 1 ˆ wmt ( ) ( ) ª log( 10 ) 9 >> 1 Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 14

15 exp(-d(t)) w << t -1 How can so slow change of the Q:? energy splitting cause any decoherence, A: if DE is practically a constant during the process of quantum oscillations. It is not a true decoherence. It is rather a temporally inhomogeneous broadening: Different shots cause oscillations with slightly different frequencies The echo technique allows one to get rid of the temporally inhomogeneous broadening: D echo ( t ) µ Ê sin Ë w 4 Ú Á w m ( wt 4) ˆ dw w This integral is determined by w of the order of t -1 Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 15

16 Decoherence factor Let the spacing between the two levels, DE, fluctuate in time: ( ) ( 0) Imax t -D t I max d e ( ) ( t ) DE - DE Gate error What is the origin of the 1/f noise? 1. One fluctuator (e.g., level system) - telegraph noise. Time-dependent dipole moment d(t): Fourier g d( 0) d( t) µ exp( - g t) transform w + g. Several fluctuators with different values of the relaxation rate g : d i ( 0) d ( t) µ Ensemble Â C i C ( ) g w w + g Ú P g d averaging w + g i i g i C i is the coupling constant of the i th fluctuator, and < C> is its mean value. P(g) is the probability density of the relaxation rates. We assumed that C i and g i are statistically independent 3. Assuming now that log(g) is distributed uniformly (it is natural if g µ exp( - S)), i.e., P ( g ) µ 1 g we obtain 1 d ( 0 ) d ( t) µ w w e g Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 16

17 B.A. & B. Z. Spivak, JETP Letters, v. 49, #8, p. 57 (1989) Fluctuations in the intensity of 1/f noise in disordered metals d i ( 0) d ( t) µ Â C w i w + i i g g i C i is the coupling constant g i is the relaxation rate d ( t ) DE - DE For a small system (qubit) each contribution,d i (t),is inverse proportional to the cube of the distance, r, between the d t µ d t r system and the fluctuating dipole moment: ( ) ( ) 3 i The decoherence is dominated by a single fluctuator. Telegraph noise 3 t tg < 1 D( t ) µ t tg > 1 1/f noise D t µ t ( ) Different fluctuators dominate at different t The decoherence is dominated by a single fluctuator. D ( t ) 3 t µ t D( t ) t tg < 1 tg > 1 Telegraph noise D ( t ) µ t 1/f noise µt t Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 17

18 Quantum oscillations in two coupled charge qubits Yu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin & J. S. Tsai 1 t coupled ª 4 t uncoupled Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 18

19 Quantum oscillations in two coupled charge qubits Yu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin & J. S. Tsai Quantum oscillations in two coupled charge qubits Yu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin & J. S. Tsai (0,0) [(1,0) + (0,0) [(0,1) is not distinguished from (0,0) [(1,1) Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 19

20 Consider a spinor (total phase is irrelevant) Ê Y = y eij ˆ Á Ë y Ø e -ij Average components of the spin can be expressed through the absolute values of the spinor s components and the phase Decoherence of a spin 1/ Sz = y - y Sx = 1- Sz cosj S y = 1- Sz sinj Ø A random time-dependent magnetic field B r ( t) changes the phase in an uncontrollable way it rotates the spin. S r This can be called decoherence However A classical magnet also gets rotated by external magnetic fields B r ( t) S=1/ S=3 S=13/ S>>1 classical limit Harmonic oscillator B r S r ( t) Just some oscillations is not exclusively a quantum effect! For the quantum computation two-level systems (S=1/) are necessary. Involvement of highly excited states is not permissible Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 0

21 < <1 a<1 + b< leakage Decoherence Inelastic Dephasing Gate errors Leakage Interaction with thermal fluctuations of the (equilibrium) environment Spontaneous relaxation of an excited quantum system with emission of excitations in the environment Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 1

22 Inelastic dephasing hw hw other electrons phonons magnons two level systems Can be modeled, e.g., by an interaction with an oscillator bath J µ E F Persistent current E is the ground state energy F J Completely quantum phenomenon!? Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03)

23 Persistent current at zero temperature is a property of the ground state! F J µ E F E is the ground state energy Interaction between electrons can change both E and J, but this does not mean that there is dephasing of the ground state wave function. Measurements of the persistent current as well as of other thermodynamic properties do not allow to extract the dephasing rate. J We prove that the ground state of a system of N fermions is to the groundstate in the presence of a finite range scattering potential, as N Æ. This implies that the responce to application of such a potential involves only emission of excitations into the continuum, and that certain processes in Fermi gases may be blocked by orthogonality in a low -T, low - energy limit. Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 3

24 D Interaction with the environment suppresses the tunneling, i.e., the splitting. Is it a dephasing? Typical problem/mistake: Quantum System interaction Environment Y = ij Y e Nobody yet invented device that can universally serve as a phasometer. Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 4

25 Quantum System I interaction Quantum System II Y I = Y I i I e j Y II = Y II i e j II Y tot = Y tot e ij tot ( Hˆ + Hˆ + Hˆ I II int) Ytot = EYtot Quantum System I interaction Quantum System II Y tot = Y tot j e i tot a a ( Hˆ + Hˆ + Hˆ ) Y = E Y I II int tot a tot Ground state: 0 Y tot Y 0 tot Y 0 I ƒ Y 0 II Y 0 I ƒ Y 0 II is not even an eigenstate of the system. It is a complex superposition of excited states. Being prepared in this particular state the system starts to evolve in time, and this evolution can look like a phase relaxation of, say, the system I. Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 5

26 Quantum System I interaction Quantum System II Ground state: 0 Y tot Y 0 tot Y 0 I ƒ Y 0 II Y 0 I Q: ƒ Y 0 II is not even an eigenstate of the system. It is a complex superposition of excited states. Being prepared in this particular state the system starts to evolve in time, and this evolution can look like a phase relaxation of, say, the system I. Does it mean that there is? zero temperature dephasing A: No Inelastic dephasing hw hw other electrons phonons magnons two level systems Can be modeled, e.g., by an interaction with an oscillator bath Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 6

27 e-e interaction Electric noise Fluctuation- dissipation theorem: Electric noise - randomly time and space - a r r a dependent electric field E (, t) E ( k, w). Correlation function of this field is completely determined by the conductivity s k r,w : ( ) E a E b = w r Ê w ˆ k k a b cothá µ ( w, k) T k s (, k) r w, k s w ab Ë ab T r Noise intensity increases with the temperature, T, and with resistance E a E b h ( L) - Thouless conductance def. e R( L) R( L) - resistance of the sample with length (1d) area (d) g = w a b cothá µ ( w, k) T k s (, k) r w, k s w ab Ë ab 1 t f r µ Ê w ˆ k k Dephasing rate due to e-e interaction for 1d and d cases T g ( L ) f T { } r L Lf Dt f - dephasing length D - diffusion constant of the electrons Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 7

28 1 t f µ T g ( L ) f Fermi liquid is valid (one particle excitations are well defined), provided that Tt f ( T) > h 1. In a purely1d chain, g 1,and, therefore, Fermi liquid theory is never valid. ( ) 1. In a multichannel wire g L f >,provided that Lf is smaller than the localization length, and Fermi liquid approach is justified 1 t f µ T g ( L ) f g ` ( L) µ L d - where is the number of dimensions: d=1 for wires; d= for films, Lf µ t f L f µ T -1 ( 4-d ) t f µ T - ( 4-d ) T µ T -1-3 d d = = 1 Dr. Boris Altshuler, KITP, Princeton, & NEC Research Inst (KITP Glassy States 4/08/03) 8

Introduction to Theory of Mesoscopic Systems

Introduction to Theory of Mesoscopic Systems Boris Altshuler Princeton University, Columbia University & NEC Laboratories America Lecture 5 Beforehand Yesterday Today Anderson Localization, Mesoscopic