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Chapter 15 Decoherence, Refocusing and Dynamical Decoupling of Spin

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Chapter 15 Decoherence, Refocusing and Dynamical Decoupling of Spin Chapter 15 Decoherence, refocusing and dynamical decoupling of spin qubits A spin degree of freedom is a robust qubit candidate due to its extremely weak coupling to external reservoirs if we compare it to other degrees of freedom such as a charge degree of freedom. However, a very fact that we can manipulate a spin qubit with an external control force indicates that a spin qubit still couples to external degrees of freedom and loses its quantum coherence with a ¯nite time. In this chapter we will study the decoherence properties of a spin qubit and several pulse techniques to eliminate such an extrinsic decoherence e®ect. 15.1 Decoherence of spin qubits 15.1.1 Transverse relaxation (T2) process [1, 2] A spin qubit loses its phase information by a fluctuating magnetic ¯eld along z-axis (quan- tization axis). The relevant interaction for this particular decoherence process is abstractly represented by the Zeeman Hamiltonian: H^ = ¡γ¹h [H0 + ¢H(t)] I^z = ¡¹h [!0 + ¢!(t)] I^z : (15.1) Here H0 is a dc magnetic ¯eld along z-axis, !0 = γH0 is the Larmor frequency of a spin, ¢H(t) is a fluctuating magnetic ¯eld along z-axis and ¢!(t) = γ¢H(t) is a corresponding frequency (modulation) noise. The origin of ¢H(t) might be a paramagnetic impurity near a spin qubit or surrounding nuclear spin bath or stray magnetic ¯eld. Suppose the initial spin state is given by 1 jÃ(0)i = p (j "i + j #i) ; (15.2) 2 the ¯nal state after a free evolution according to (15.1) is µ ¶ R t R t 1 ¡i !0 t ¡ i ¢!(t0)dt0 i !0 t i ¢!(t0)dt0 jÃ(t)i = p e 2 e 2 0 j "i + e 2 e 2 0 j #i ; (15.3) 2 1 If we move from a laboratory frame to a rotating frame, in which x ¡ y axis rotates along z-axis with an angular frequency !0, (15.3) can be expressed as ³ ´ 1 ¡ 1 ¢Á(t) 1 ¢Á(t) jÃ(t)i = p e 2 j "i + e 2 j #i ; (15.4) 2 where Z t ¢Á(t) = ¢!(t0)dt0 : (15.5) 0 If an instantaneous Larmor frequency noise ¢!(t) has an in¯nitesimally short correla- tion time, its noise spectrum S¢!(­) becomes a white noise and the resulting phase noise given by (15.5) features a non-stationary random walk di®usion as shown in Fig. 15.1. This is a Wiener-Levy process studied in Chapter 1. Figure 15.1: A random walk of a phase ¢Á(t) of a spin qubit jÃ(t)i due to a fluctuating magnetic ¯eld ¢H(t). By introducing a gated function between [0;T ], we can employ the Fourier analysis. A co-variance function for the phase noise is de¯ned by Z t+¿ Z t h¢Á(t + ¿)¢Á(t)i = h¢!(t0)¢!(t")idt0dt" : (15.6) 0 0 Since an instaneous frequency noise ¢!(t) is a statistically stationary and ergodic process, we can replace the ensemble average in the right hand side of (15.6) with the time average, Z T 1 2 h¢!(t0)¢!(t")i = lim ¢!(t + ¿)¢!(t)dt (15.7) T !1 T ¡ T Z 2 1 1 = S¢!(­) cos(­¿)d­ : 2¼ 0 Here ¿ = t0 ¡ t" and we used the Wiener-Khintchine theorem. From (15.6) and (15.7), we have Z 1 Z t+¿ Z t 1 0 £ 0 ¤ h¢Á(t + ¿)¢Á(t)i = d­S¢!(­) dt dt" cos ­(t ¡ t") (15.8) 2¼ 0 0 0 2 Z 1 1 1 = d­S¢!(­) 2 f1 + cos(­¿) ¡ cos(­t) ¡ cos [­(t + ¿)]g : 2¼ 0 ­ By setting ¿ = 0, we can obtain the variance of the phase noise: Z 1 2 1 1 h¢Á(t) i = S¢!(­ » 0) d­ 2 [1 ¡ cos(­t)] ; (15.9) ¼ 0 ­ 1 = S (­ » 0)t ; 2 ¢! where we use the fact that S¢!(­) is a white noise so that S¢!(­) can be replaced by its zero-frequency spectral density S¢!(­ ' 0). If we introduce a phase di®usion constant D¢Á by 2 h¢Á(t) i = 2D¢Át ; (15.10) the phase di®usion constant is uniquely determined by the zero frequency spectral density S¢!(­ ' 0) of an instantaneous frequency, 1 D = S (­ ' 0) : (15.11) ¢Á 4 ¢! The 2 £ 2 density matrix of the initial spin state (15.2) is expressed as à ! à ! ½ ½ 1 1 1 ½^ = "" "# = : (15.12) ½#" ½## 2 1 1 The instantaneous frequency noise ¢!(t) does not decay the diagonal terms ½"" and ½## but decays the o®-diagonal terms, 1 i¢Á(t) ½"#(t) = ½#"(t) = he i (15.13) 2 ½ ¾ 1 1 = exp ¡ h¢Á(t)2i 2 2 1 = exp [¡D (t)] : 2 ¢Á 1 4 The o®-diagonal term decays exponentially with a time constant T2 = = . D¢Á S¢!(­'0) Since the instantaneous frequency noise ¢!(t) is proportional to the magnetic ¯eld fluc- tuation ¢H(t), the zero-frequency spectral density of ¢H(t) ultimately determines the magnitude of a phase coherence time T2. Remark: 1. In real experimental situations, the instantaneous frequency noise ¢!(t) has a vary- ing spectral shape depending on magnetic environments. Often, 1=f noise dominates at low frequencies due to the distributed random telegraphic signals as discussed in chapter 9. At high frequencies, quantum zero-point fluctuations proportion to fre- quency ! always dominate, as discussed in chapter 4. We must evaluate Z 1 2 1 1 h¢Á(t) i = d­S¢!(­) 2 [1 ¡ cos(­t)] ; (15.14) ¼ 0 ­ instead of (15.9) to calculate the decay of the o®-diagonal elements ½"# and ½#". 3 2. An applied dc ¯eld H0 is not completely uniform in all space points. If many spin qubits are placed in such an inhomogeneous dc ¯eld, they have di®erent Larmor frequencies. This leads to the dephasing e®ect if we compare the phase di®erence between di®erent qubits. A time constant for this dephasing process is determined by the spatial (not temporal) inhomogeneous broadening of the dc ¯eld and distin- ¤ guished from T2 process. A new time constant is often referred to as T2 . 15.1.2 Longitudinal relaxation (T1) process [3] A spin qubit loses its amplitude information by a fluctuating transverse magnetic ¯eld in a x ¡ y plane. The relevant interaction for this particular spin relaxation process is abstractly represented by the spin-boson Hamiltonian: X ¡ ¢ X ³ ´ ^ ^ + ^ ^ + H = ¡γ¹hH0Iz + ¹h!s a^s a^s + 1=2 ¡ ¹hgs I+a^s + I¡a^s ; (15.15) s s + wherea ^s (^as ) is the boson annihilation (creation) operator which represent a transverse ac magnetic ¯eld. The third term represents the spin flip process by absorbing or emitting one photon. Suppose the initial spin state is given by jÃ(0)i = j "isj0if ; (15.16) where the spin is in an excited state j "is and the ¯eld is a vacuum state j0if . The ¯nal state is of the form !0 X !0 ¡i t i( ¡!s)t jÃ(t)i = C"0(t)e 2 j "; 0i + C#s(t)e 2 j #; si ; (15.17) s where j #; si represents a state in which the spin is in a ground state j #is and the ¯eld mode s acquires one photon j1is. Note that the ¯eld modes have continuous spectrum so that all the other modes except for a particular mode s remain vacuum states. The ¯nal state (15.17) should satisfy the SchÄodingerequation, d i¹h jÃ(t)i = Hj^ Ã(t)i : (15.18) dt By substituting (15.17) into (15.18) and projecting h"; 0j and h#; sj on both sides of the resulting equation, we obtain the coupled mode equations: X ¡i(!s¡!0)t C_ "0(t) = ¡i gse C#s(t) ; (15.19) s i(!s¡!0)t C_ #s(t) = ¡igse C"0(t) : (15.20) The formal integration of (15.20) with an initial condition of C#s(0) = 0 results in Z t 0 0 i(!s¡!0)t 0 C#s(t) = ¡igs dt e C"0(t ) : (15.21) 0 4 If we substitute (15.21) into (15.19), we have the following integro-di®erential equation for C (t): "0 Z X t 0 _ 2 0 ¡i(!s¡!0)(t¡t ) 0 C"0(t) = gs dt e C"0(t ) : (15.22) s 0 P The summation over all ¯eld modes s can be replaced by the integral with the energy density of states in a reasonably large volume, X Z 1 2 2 gs ! d!sg (!s) ½ (!s) : (15.23) s 0 0 In a very short time scale, we can safely assume C"0(t ) in (15.22) satis¯es 0 C"0(t ) ' C"0(0) = 1 : (15.24) Then, (15.22) is reduced to Z Z 1 t 0 2 0 ¡i(!s¡!0)(t¡t ) C_ "0(t) = = ¡ d!sg (!s) ½ (!s) dt e (15.25) 0 0 Z 1 · µ ¶¸ 3 i = ¡ d!sg (!s) ½ (!s) ¼± (!s ¡ !0) ¡ P 0 !s ¡ !0 ¡ = ¡ ¡ i±! ; 2 where 2 ¡ = 2¼g (!0) ½ (!0) ; (15.26) "Z # 1 g (! )2 ½ (! ) ±! = P s s d! ; (15.27) 0 !s ¡ !0 where P stands for a principle value. Thus, the probability amplitude for ¯nding the initial state is given by µ ¶ ¡ C (t) = 1 ¡ + i±! t : (15.28) "0 2 ¡ is the Fermi golden rule decay rate of the excited state and ±! is a frequency shift. 0 In a very long time scale, we cannot replace C"0(t ) by C"0(0) but we can still ap- 0 proximate C"0(t ) by C"0(t) in (15.22).
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