Dynamical Decoupling Noise Spectroscopy

Dynamical decoupling noise spectroscopy

Gonzalo A. Álvarez∗ and Dieter Suter† Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany.

Decoherence is one of the most important obstacles that must be overcome in quantum information processing. It depends on the qubit-environment coupling strength, but also on the spectral composition of the noise generated by the environment. If the spectral density is known, ﬁghting the effect of decoherence can be made more effective. Applying sequences of inversion pulses to the qubit system, we generate effective ﬁlter functions that probe the environmental spectral density. Comparing different pulse sequences, we recover the complete spectral density function and distinguish different contributions to the overall decoherence.

PACS numbers: 03.65.Yz, 76.60.Es, 76.60.Lz, 03.67.Pp

Introduction.— Quantum information processing relies tral distribution. In the following, we describe an exact and on the robust control of quantum systems. A quantum sys- simple method for obtaining general spectral density func- tem is always influenced by external degrees of freedom, tions that extends recent approximate solutions that can be the environment, that disturb the quantum information by used only for specific cases [10, 25]. a process called decoherence [1]. Many strategies were A qubit as the noise probe.— We consider a single qubit developed to fight this degradation of information. These Sˆ as the probe. It is coupled to the bath to be studied methods are based on correction of errors [2,3] and decou- with a purely dephasing interaction. In a resonantly ro- pling the environment [4–7]. Fighting decoherence suc- tating frame of reference [6], the free evolution Hamilto- cessfully requires knowledge of the noise spectral density nian is Hbf = HbSE + HbE, where HbE is the environment to design suitable quantum processes [8–11]. ˆ ˆ Hamiltonian and HbSE = bSESzE is a general pure de- One simple decoupling strategy is called dynamical dephasing interaction between system and environment. Eˆ is coupling (DD) [7, 12]. It is based on the application of some operator of the environment and bSE the SE coupling a sequence of control pulses to the system to effectively strength. This type of interaction is encountered in a wide isolate it from the environment. Different DD sequences range of solid-state spin systems, as for example nuclear were developed [7, 12–14] and tested experimentally [15– spin systems in NMR [4,5, 17, 19], electron spins in dia- 20]. Different sources modify the performance of DD se- monds [18], electron spins in quantum dots [27], donors in quences. Once pulse errors are small [17, 18, 20], the silicon [28], etc. spectral density of the system-environment (SE) interac- We consider the application of a sequence of short, tion becomes the main factor [8, 14, 15, 19, 21–24]. Con- strong pulses that invert the probe qubit [4,5,7, 12]. We sequently a DD sequence has to be judiciously designed assume N instantaneous pulses at times t , with delays according to the particular noise spectral density to be de- i τ = t − t between the pulses for i = 2, .., N + 1 coupled [14, 15, 21–24]. The indefinite number of possi- i i i−1 and τ = t − t , where t = 0 and t = τ . bilities for designing DD sequences leads to the imposi- 1 1 0 0 N+1 c bility of a “brute force” exploration and therefore the de- While such a sequence can refocus a static system- velopment of noise spectroscopy methodogies is required environment interaction completely, any time-dependence [10, 11, 25, 26]. reduces its efficiency. We calculate the remaining decay rate for the case where the environment can be well In this paper, we present a method to determine the spec- described by a stochastic noise. This results are also

arXiv:1106.3463v2 [quant-ph] 17 Sep 2011 tral density of the SE interaction. The method is based on valid for a quantum second order approximation of the previous results that the decay rate of a qubit during DD time-dependent SE interaction [6]. We now eliminate the is given by the overlap of the bath spectral density func- environment-Hamiltonian HbE by using an interaction rep- tion and a filter function generated by the DD sequence resentation with respect to the evolution of the isolated en- [14, 15, 19, 21–24]. The filter function is given by the vironment. The system-environment Hamiltonian then be- (E) Fourier transform of the SE interaction modified by the ˆ −iHbE t ˆ iHbE t comes Hb (t) = bSESze Ee . Since HbE does control pulses: each π-pulse changes the sign of the SE SE not commute with HbSE, the effective system-environment coupling. When many DD cycles are applied to the system, (E) the filter functions become a sum of δ−functions [19] and interaction HbSE is time-dependent and the system experi- the decoherence time is given by a discrete sum of spec- ences a fluctuating coupling with the environment. Trac- −iHbE t ˆ iHbE t tral densities. A judicious choice of the DD sequence thus ing over the bath variables replaces bSEe Ee by allows one to probe the environmental spectral density at the stochastic function bSEE(t). For simplicity we assume selected frequencies. Combining several measurements, it that this random field has a Gaussian distribution with zero is possible to obtain a detailed picture of the noise spec- average, hE(t)i = 0. The auto-correlation function is 2 hE(t)E(t + τ)i = g (τ) and the spectral density S(ω) of only the k = 1 term is a rough approximation. The of the system-bath interaction is the Fourier transform of main difficulty here is that a single measurement depends 2 bSEg (τ). on an infinite number of unknown spectral density values. The free evolution operator for a given realization of We solve this problem by a two-step procedure: in the first n ˆ o step, we combine m measurements with different pulse de- the random noise is exp −iφ(t)Sz , where φ(t) = lays, which we choose such that they probe the spectral R t bSE 0 dt1E(t1) is the phase accumulated by the probe density function at a discrete set of harmonic frequencies spin during the evolution. Considering now the effect with different sensitivity amplitudes Ak. In this step, we of the pulses, they generate reversals of HbSE(t). If the neglect contributions from the tail of S(ω > mωmin). pulses are applied during the interval τc as described above, This yields a square matrix that we can invert to obtain the the accumulated phase φ(Mτc) after M cycles becomes values of S(jωmin), j = 1..m. From the resulting spec- R Mτc 0 0 0 tral density function, we estimate a functional form for the φ(Mτc) = bSE 0 dt fN (t , Mτc)E(t ), where the 0 modulating function fN (τ , Mτc) switches between ±1 at tail of the distribution and correct the data for the contri- the position of every pulse [21]. If the initial state of the butions from the tail. Inverting the matrix again, with the ˆ corrected values, gives the final spectral density distribu- probe spin is ρˆ0 = Sx,y, its normalized magnetization under the effects of DD taking the average over the random tion. 1 2 A natural choice for the probing sequence is the CPMG − 2 hφ (t)i fluctuations is hsx,y(t)i = e , and its decay can be or equidistant sequence, which has harmonics at frequen- quantified by the exponential’s argument cies ω = π/τ. To simplify inverting equation (2), we Z ∞ 0 1 2 q π 2 choose the pulse delays in the different measurements such 2 φ (t) = R(t) t = 2 dωS(ω) |FN (ω, Mτc)| , −∞ that all relevant frequencies, including all harmonics, are (1) multiples of a minimal frequency ωmin. We therefore start where FN (ω, Mτc) is the Fourier transform of with a maximum delay τmax = π/ωmin, which deter- 0 fN (t , Mτc) [21]. The decay function R(t) t is thus mines the frequency resolution with which we probe the equal to the product of the spectral density S(ω) of spectral density function. If the maximum frequency at the system-environment coupling and the filter transfer which we want to probe the spectral density function di- function FN (ω, Mτc). We have recently shown that rectly is mωmin, then we need to apply sequences with considering the infinite extension of the modulating delays τn = τmax/n = τminm/n. If we neglect the con- 0 function, fN (t , ∞), by a convolution this provides a tribution from frequencies > mωmin, the relaxation rates FN (ω, Mτc) that is a sum of sinc functions centered at Rn for the different experiments are given by a system of the harmonic frequencies kω0 = 2πk/τc of the Fourier m linear equations 0 series of fN (t , ∞) [19]. Hence for t = Mτc τB, the noise correlation time, the filter function |FN (ω, τM )| [m/n] m X 2 X becomes an almost discrete spectrum given by the Fourier Rn = AkS(nkωmin) = UnjSj, (3) 0 transform of fN (t , ∞), i.e. F (ω, t) is represented by k=1 j=1 a series of δ-functions centered at kω0 neglecting the where [m/n] denotes the integer part of m/n and j = nk. 2 contributions from the secondary maxima. Thus, in the The elements Ak form an upper triangular matrix Unj = limit of many cycles, the decay is exponential and R(t) P[m/n] 2 A δj,nk, and Sj = S(jωmin) represent the un- becomes a constant k=1 k ∞ known spectral density values, which can formally be cal- Pm −1 X 2 culated as S = (U ) R R(t) = R = A S (kω ) , (2) j n=1 jn n k 0 We now correct for the omitted contributions from the k=1 √ high-frequency tail of the infinite sum by approximating it 2 2π 2 with Ak = 2 |FN (kω0, τc)| , where for a CPMG se- τc with a suitable functional form, which depends on the sys- quence [4,5] with τ2 = 2τ1 = 2τ3 = τ, Ak ∝ 1/k for tem being studied. Typical examples include a power law odd k and 0 otherwise. This is the basis for the DD noise decay, lorentzian or gaussian decay, or a sudden cut off like spectroscopy methodology presented in this letter. Exam- in an ohmic bath. In the system that we used as an examples of the probe spin signal decay are shown in the inset ple (see below), the experimental data can be approximated of Fig.1. very well by a power law dependence, as shown in Fig.1 Noise spectroscopy.— Assuming for the moment that the (squares). C sum in Eq. (2) collapses to the k = 1 term, we can clearly If the tail satisfies a power law Sj = jα for j > np, then trace out the bath spectral density by varying the delay be- ∞ 2 ∞ 2 X A C C X A CΛα tween the pulses as in Ref. [10, 25]. However, for real R = k = k = . (4) n>np (nk)α nα kα nα DD sequences, we always have an infinite series, where k=1 k=1 all harmonics contribute to the decay rate with the weight This relation is represented by the black solid line in Fig.1. Ak. Determining the spectral density function therefore re- We can now modify Eq. (3) by adding the neglected terms quires the inversion of Eq. (2) and thus the consideration and then the relaxation rates satisfy 3

8 1 13 0 Fig.1 shows examples of the C signal decays. The lines 100 -3.59±0. in the inset show the ﬁtted exponential decays, which agree 13 13  C- C Relaxation ~ Signal very well with the data points in this range. This demon- CPMG delays s) (ms) 0.1 71.21 97.56 -1 strates that we are in the regime where the ﬁlter functions 79.59 106.53 10 87.52 121.71 are discrete. KDD was shown to be robust against pulse

0 10203040 errors, independent of the initial condition (please see Ref. CPMG Evolution time M (ms) KDD c [20] for details). For ideal pulses, both sequences have the 1 same ﬁlter function. As shown in Fig.1 (squares) the observed relaxation times for this system depend on the pulse Free evolution decay time spacing like ∝ τ −3.59 for the CPMG sequence over the

Relaxation time R  0.1 B range τ =[50 µs, 110 µs]. We only used the data points 0.1 1 for τ > 50 µs to determine the parameters C and α, since Pulse delay  (ms) Fig.1 indicates that other processes contribute to the relaxation at shorter delays. From the fitting process, we found Figure 1: (Color online) Experimental relaxation times of the α = 3.59 ± 0.08 and Λα ≈ 1.002. Here, the contribution probe spin under the application of CPMG (squares) and KDD of the infinite series of 2·10−3 is almost negligible. Figure (circles) sequences. The black solid line represents a power law 1 also shows that the dependence of the decoherence rates fitting to the CPMG data and the green dotted line the asymptotic changes at τ 100 µs. This agrees with the value that we free evolution decay rate. The black dashed line is a fitting to the & 0 α −1 determined earlier for the correlation time of the bath τB KDD data with an expression (R13C + C τ ) . Inset: Experi- mental signal decays of the probe spin as a function of the evolu- [17]. tion time under CPMG dynamical decoupling. Different curves We observe in the KDD case that the relaxation time sat- correspond to different pulse delays. The straight lines represent urates for τ shorter than 50 µs and in general is lower than exponential fits. the CPMG cases (Fig.1, circles). This difference can be attributed to the effect of 13C-13C couplings. Because in the CPMG sequence all pulses generate the same rotation, the overall effect of the pulse cycle is to first order equivalent m m ! Λ C C to a constant effective field, which stabilizes the observable X α X 13 13 Rn = UnjSj + − Unj , (5) magnetization against the effect of C- C couplings [29]. nα jα j=1 j=1 In the KDD case, the state is not longitudinal to the pulses 2 and no spin-lock effect is observed. The saturation of the ΛαC Pm C C P Ak where nα − j=1 Unj jα = nα k>m−n+1 kα rep- relaxation time for the CPMG case for τ < 50 µs can be resents the effective spectral density summing the con- attributed to the finite rf field strength or, equivalently, to tribution from all harmonics k > m − n + 1. the finite duration of the pulses. Pulse errors may also con- The spectral density is now determined from Sj = tribute in this regime. Pm −1 ΛαC C 13 13 n=1 (U )jn Rn − nα + jα . Eq. (4) shows that for a In the KDD case, the C- C interaction eclipses the power law dependence, the relaxation rate and the spectral spectral density of the proton bath. To verify this, we as- density are proportional and thus for a qualitative descrip- sumed a 13C-13C relaxation rate independent of the pulse 0 α −1 tion of S(ω) considering only the k = 1 term is enough delays and fitted the expression (R13C + C τ ) over validating the results of Ref. [10, 25]. the range [50 µs, 110 µs] where the CPMG data follow a Experimental determination of S(ω).— For an experi- power law (dashed line in Fig.1). We obtained R13C = mental demonstration of this method, we chose 13C nuclear (75 ± 1) s−1 for the 13C-13C relaxation rate and α = spins (S = 1/2) as probe qubits. We used polycrystalline 3.7 ± 0.3, which perfectly matches with the CPMG result. adamantane where the carbon nuclear spins are coupled to If we subtract the R13C contribution, we obtain the spectral an environment of 1H nuclear spins (I = 1/2) that act as density represented by the empty circles in Fig.2, which a spin-bath. The natural abundance of the 13C nuclei is are almost identical to the result obtained with the CPMG about 1%, and to a good approximation each 13C nuclear sequence (solid squares). spin is surrounded by about 150 1H nuclear spins. The We demonstrate with Eq. (4) that the qualitative be- interaction with the environment is thus dominated by the haviour of the power law tail can be well obtained by the 13C-1H magnetic dipole coupling [6]. To determine the first harmonic approximation (k = 1) that is proportional bath spectral density we applied the equidistant sequences to the exact solution derived from (5). For lower frequen- CPMG and KDD [20] to the probe spin for different de- cies, the corrections from our exact method can be relevant lays between pulses τn = τmax/n, with n = 1..40 and (squares vs. rombuses in Fig.2). However, because S(ω) τmax = 2ms. Delays are measured between the center of decays rapidly, the difference is small in this case. the pulses. For CPMG, we chose an initial state longitu- This is not always true, as we now show with a spe- dinal to the rf field of the refocusing pulses because then cific example: We modulate the system-environment inter- pulse error effects can be neglected [5, 17]. The inset of action by applying a resonant radio-frequency field to the 4

1.92 3.85 7.69

S(0) -3.59±0.08 1.0 =0 ~ =1.92 kHz ) (kHz) 1 4

 =3.85 kHz =7.69 kHz ) (kHz) ) (kHz)

 2 0.8  S(

0.0 0.5 1.0 (2) (KHz) 0.6 13C-13C Relaxation rate R 0.1 13C CPMG KDD 0.4 CPMG (1st harmonic) 13 13 Spectral density S( KDD ( C- C relaxation substracted) 0.2 0.01 1 Spectral density S( Frequency (2) (kHz) 0.0 110 Frequency (2) (kHz) Figure 2: (Color online) Experimentally determined noise spectral density. The inset shows in a linear scale the low frequency Figure 3: (Color online) Experimentally determined noise spec- regime. The green dotted line represents the free evolution decay tral density for a modulated system-environment interaction. The rate of the probe spin, i.e. S(0). empty symbols were obtained when only the first harmonic is considered, the full symbols show the results from the exact method. proton spins, which periodically inverts them. Using the same measurement procedure, we obtained the data shown in Fig.3. dard NMR techniques that use CPMG sequences to distin- 2 The approximate solution S(nωmin) = Rn/A1, where guish between difference sources of inhomogenities [30] only the first harmonic (k = 1) is considered, shows a or measuring diffusion rates [31–33] as well as protein dy- main peak at the modulation frequency Ω and some satel- namics rates [34] in liquid state NMR. Those methods de- lite peaks at at lower frequencies that are integer fractions termine correlation times but assume specific spectral den- of Ω. These satellite peaks are artefacts of the data analy- sity functions, while our technique is suitable for the deter- sis that neglects contributions from higher harmonics of the mination of unknown spectral densities. filter function of Eq. (2). Using our method, we obtain an Acknowledgments.— This work is supported by the improved solution where the satellite peaks are eliminated. DFG through Su 192/24-1. GAA thanks financial support The spectral density distribution for this case is qualita- from the Alexander von Humboldt Foundation in the ini- tively different from that of the unmodulated case. In par- tial stage of this project. We thank Alexandre M. Souza for ticular, the value at zero frequency is reduced, but a max- helpful discussions. imum has appeared at the modulation frequency. This has important consequences for implementing dynamical decoupling: A good decoupling performance when Ω = 7.69 kHz is expected for τ ∼ 0.12 ms, where the first harmonic ∗ [email protected] is at the spectral density minimum near 4 kHz. Increas- † ing the decoupling rate then would drastically reduce the [email protected] decoupling performance, in stark contrast to the usual ex- [1] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003). [2] J. Preskill, Proc. R. Soc. Lond. A 454, 385 (1998). pectation that it should increase with the decoupling rate. [3] E. Knill, Nature 434, 39 (2005). Conclusions.— We have developed a method to deter- [4] H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954). mine the noise spectral density generated by a bath. It is [5] S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958). based on modulating the system-environment interaction [6] A. Abragam, Principles of Nuclear Magnetism (Oxford by applying sequences of inversion pulses to the system. If University Press, London, 1961). the sequence consists of many repetitions of a basic cy- [7] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 cle, the resulting decays are exponentials and the decay (1999). rates are given by the spectral density at discrete frequen- [8] J. Zhang, X. Peng, N. Rajendran, and D. Suter, Phys. Rev. cies. This allows one to build a linear system of equations A 75, 042314 (2007). that can be inverted to obtain the unknown spectral den- [9] D. D. Bhaktavatsala Rao and G. Kurizki, Phys. Rev. A 83, 032105 (2011). sity function. We applied the method to obtain the spectral 13 1 [10] J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, density of the C- H interaction in adamantane. Apply- G. Fitch, D. G. Cory, Y. Nakamura, J. Tsai, and W. D. ing this method to other systems will help fighting deco- Oliver, Nat Phys 7, 565 (2011). herence, e.g. by optimizing DD sequences by reducing the [11] I. Almog, Y. Sagi, G. Gordon, G. Bensky, G. Kurizki, and overlap of their filter functions with the noise spectral den- N. Davidson, J. Phys. B: At., Mol. Opt. Phys. 44, 154006 sity [14, 15, 21–24]. Our method also complements stan- (2011). 5

[12] W. Yang, Z. Wang, and R. Liu, Front. Phys. 6, 2 (2010). 103, 040501 (2009). [13] K. Khodjasteh and D. A. Lidar, Phys. Rev. Lett. 95, 180501 [24] J. Clausen, G. Bensky, and G. Kurizki, Phys. Rev. Lett. 104, (2005). 040401 (2010). [14] G. S. Uhrig, New J. Phys. 10, 083024 (2008). [25] C. A. Meriles, L. Jiang, G. Goldstein, J. S. Hodges, J. Maze, [15] M. J. Biercuk, et al., Nature 458, 996 (2009). M. D. Lukin, and P. Cappellaro, J. Chem. Phys. 133, 124105 [16] J. Du, et al., Nature 461, 1265 (2009). (2010). [17] G. A. Álvarez, A. Ajoy, X. Peng, and D. Suter, Phys. Rev. [26] K. C. Young and K. B. Whaley, arXiv:1102.5115 (2011). A 82, 042306 (2010). [27] R. Hanson, et al., Rev. Mod. Phys. 79, 1217 (2007). [18] C. A. Ryan, J. S. Hodges, and D. G. Cory, Phys. Rev. Lett. [28] B. E. Kane, Nature 393, 133 (1998). 105, 200402 (2010). [29] G. E. Santyr, R. M. Henkelman, and M. J. Bronskill, J. [19] A. Ajoy, G. A. Álvarez, and D. Suter, Phys. Rev. A 83, Magn. Reson. 79, 28 (1988). 032303 (2011). [30] A. N. Garroway, J. Magn. Reson. 28, 365 (1977). [20] A. M. Souza, G. A. Álvarez, and D. Suter, Phys. Rev. Lett. [31] E. O. Stejskal, J. Chem. Phys. 43, 3597 (1965). 106, 240501 (2011). [32] E. O. Stejskal and J. E. Tanner, J. Chem. Phys. 42, 288 [21] L. Cywinski, R. M. Lutchyn, C. P. Nave, and S. DasSarma, (1965). Phys. Rev. B 77, 174509 (2008). [33] K. J. Packer, J. Magn. Reson. 9, 438 (1973). [22] G. Gordon, G. Kurizki, and D. A. Lidar, Phys. Rev. Lett. [34] A. Mittermaier and L. E. Kay, Science 312, 224 (2006). 101, 010403 (2008). [23] H. Uys, M. J. Biercuk, and J. J. Bollinger, Phys. Rev. Lett.