Spin-echo of a single electron spin in a quantum dot
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Spin-echo of a single electron spin in a quantum dot
F.H.L. Koppens, K.C. Nowack, and L.M.K. Vandersypen Kavli Institute of NanoScience Delft, P.O. Box 5046, 2600 GA Delft, The Netherlands
We report a measurement of the spin-echo decay of a single electron spin confined in a semicon-ductor quantum dot. When we tip the spin in the transverse plane via a magnetic field burst, itdephases in 37 ns due to the Larmor precession around a random effective field from the nuclearspins in the host material. We reverse this dephasing to a large extent via a spin-echo pulse, andfind a spin-echo decay time of about 0.5 µ s at 70 mT. These results are in the range of theoreticalpredictions of the electron spin coherence time governed by the dynamics of the electron-nuclearsystem. Isolated electron spins in a semiconductor can havevery long coherence times, which permits studies of theirfundamental quantum mechanical behavior, and holdspromise for quantum information processing applications[1]. For ensembles of isolated spins, however, the slow in-trinsic decoherence is usually obscured by a much fastersystematic dephasing due to inhomogeneous broadening[2, 3]. The actual coherence time must then be esti-mated using a spin echo pulse that reverses the fastdephasing[4]. In this way, decoherence times as long ashundreds of microseconds have been measured for en-sembles of electron spins bound to phosphorous donorsin silicon [5].For a single isolated spin there is no inhomogeneousbroadening due to averaging over a spatial ensemble. In-stead, temporal averaging is needed in order to collectsufficient statistics to characterize the spin dynamics. Insome cases, this averaging can also lead to fast apparentdephasing that can be (largely) reversed using a spin-echo technique. This is possible when the dominant in-fluence on the electron spin coherence fluctuates slowlycompared to the electron spin dynamics, but fast com-pared to the required averaging time. Such a situationis common for an electron spin in a GaAs quantum dotwhere the hyperfine interaction with the nuclear spinsgives rise to an effective slowly fluctuating nuclear field,resulting in a dephasing time of about tens of nanosec-onds [6, 7, 8, 9, 10]. The effect of the low-frequency com-ponents of the nuclear field can be reversed to a largeextent by a spin-echo technique. For two-electron spinstates, this was demonstrated by rapid control over theexchange interaction between the spins [10]. The appli-cation of a spin-echo technique on a single electron spin isrequired when using the spin in a GaAs quantum dot asa qubit. Furthermore, erasing fast dephasing allows fora more detailed study on the remaining decoherence pro-cesses such as the rich dynamics of the electron-nuclearsystem [11, 12, 13, 14].Here, we report the use of a spin-echo technique forprobing the coherence of a single electron spin confinedin an electrostatically defined GaAs quantum dot (shownin Fig. 1a). The spin is manipulated using electron spinresonance, as reported in [15]. We first realize a two-pulse experiment akin to Ramsey interference, whereby fringesdevelop when the relative phase between the pulses is var-ied. By varying the delay time between the pulses, wemeasure a dephasing time T ∗ of about 37 ns. When us-ing a spin-echo technique, a spin-echo decay time T , echo is obtained of about 0.5 µ s. This is more than a factor often longer than the Ramsey decay time, indicating thatthe echo pulse reverses the dephasing to a large extent.These findings are consistent with theoretical predictionsfor this system [6, 7, 11, 12, 13, 16], as well as with earlierecho measurements on two-electron spin states in a simi-lar quantum dot system [10], and with mode locking mea-surements of single spins in an ensemble of self-assembledquantum dots [17].The measurement scheme is depicted in Fig. 1a. Twoquantum dots are tuned such that one electron alwaysresides in the right dot and a second electron can flowthrough the two quantum dots only if the spins are anti- V P -D V P aD V P RF a P I dot B ac B ext m m I CPS b odd m s FIG. 1: (color online) a) Bottom: Scanning electron mi-croscope (SEM) image of the Ti/Au gates on top of aGaAs/AlGaAs heterostructure containing a two-dimensionalelectron gas 90 nm below the surface. White arrows indicatecurrent flow through the two coupled dots (dotted circles).The gate labeled with V p is connected to a homemade bias-tee(rise time 150 ps) to allow fast pulsing of the dot levels. Top:SEM image of the on-chip coplanar stripline, separated fromthe surface gates by a 100-nm-thick dielectric. Due to thegeometry of the stripline, the oscillating field with amplitude B ac and frequency f ac is generated primarily perpendicular tothe static field B ext , which is applied in the plane of the two-dimensional electron gas. b) Schematic of the electron cycle(time axis not on scale). The voltage ∆ V p (with lever arm α )on the gate detunes the dot levels during the manipulationstage (applied bias is 1.5 mV).
30 ns 90 ns t (ns) B =42 mT ext p /2 /2 a b c t (ns) B /2 s ac t t (ns) wait D o t c u rr en t ( f A ) -2 p - p p p Phase 2nd pulse D o t c u rr en t ( f A ) T ~(37±5) ns * Burst time (ns) I ( f A ) FIG. 2: (color online) a) Ramsey signal as a function of free-evolution time τ . Each data point reflects a current mea-surement averaged over 20 seconds at constant B ext =42 mT, f ac = 210 MHz, B ac = 3 mT (as shown in the inset, this givesa Rabi period τ π of 120 ns; see [15] for further details). Inorder to optimize the visibility of the decay, the second pulseis a 3 π/ π/ P odd , see main text). Solid line: Gaus-sian decay with T ∗ = 30 ns, corresponding to σ = 1 . P odd iscomputed taking σ = 1 . I dot = P odd ( m + 1)80 + 23 fA ( m and offset due tobackground current obtained from fit). A current of 80 fAcorresponds to one electron transition per 2 µs cycle, and m is the additional number of electrons that tunnels through thedot on average before the current is blocked again. Here, wefind m = 1 .
44; the deviation from the expected m = 1 is notunderstood and discussed in [15]. b) Measured and numer-ically calculated Ramsey signal for a wide range of drivingfields. The simulations assume σ = 1 . P odd ( m + 1)80 + 23 fA ( m = 1 .
5) for τ π =40-220ns, and as P odd ( m + 1)80 + 43 fA ( m = 1 .
5) for τ π =440 ns.c) Ramsey signal as a function of the relative phase betweenthe two RF bursts for τ = 10 (crosses) and 150 ns (circles).Gray dashed line is a best fit of a cosine to the data. parallel. For parallel spins, the second electron cannotenter the right dot due to the Pauli exclusion principle,and is blocked in the left dot [18]. This allows us toinitialize the system in a mixed state of |↑↑i and |↓↓i (stage 1), although from now on, we assume the initialstate is |↑↑i , without loss of generality. Next the electronspins are manipulated with a sequence of RF bursts [19](stage 2), while a voltage pulse ∆ V p is applied to one ofthe gates so that tunneling is prohibited regardless of the spin states. Once the pulse is removed, electron tunnelingis allowed again, but only for anti-parallel spins (stages3 and 4). The entire cycle lasts 2 µ s and is continuouslyrepeated, resulting in a current flow which reflects theaverage probability P odd to find anti-parallel spins at theend of stage 2.We first use this scheme to measure the dephasing ofthe spin via a Ramsey-style experiment (see inset Fig. 2afor the RF pulse sequence). A π/ |↑i and |↓i , afterwhich the spin is allowed to freely evolve for a delay time τ (for now, we reason just in a single-spin picture [20]).Subsequently, a 3 π/ |↑i , and the system returns to spinblockade. If the phases of the two pulses are 180 ◦ out ofphase, the spin is instead rotated to |↓i , and the blockadeis lifted. Fig. 2c shows that for small τ , the probabilityto find |↓i indeed oscillates sinusoidally as a function ofthe relative phase between the two RF pulses. This isanalogous to the well-known Ramsey interference fringes.For large τ , however, the spin completely dephases duringthe delay time, and the fringes disappear (Fig. 2c). Wesee that the maximum contrast for the effect of dephasingis obtained for two pulses with the same phase. Thesignal for this case is shown in Fig. 2a, as a function of τ .We find that the signal saturates on a timescale of T ∗ ∼
37 ns (obtained from a Gaussian fit, see below), whichgives a measure of the dephasing time.The observed Ramsey decay time is the result of thehyperfine interaction between the electron spin and therandomly oriented nuclear spins in the host material.The coupling Hamiltonian is given by H hf = S · h = S z h z + ( S + h − + S − h + ) where S is the spin-1/2 op-erator for the electron spin and h = P i A i I i , with I i the operator for nuclear spin i and A i the correspond-ing hyperfine coupling constant. The S z h z term in theHamiltonian can be seen as a nuclear field B N,z in the z-direction that modifies the Larmor precession frequencyof the electron spin (the other two terms are discussedbelow in the context of spin-echo). This nuclear fieldfluctuates in time with a Gaussian distribution withwidth σ and a typical correlation time ∼ µs − s .This is much longer than the 2 µs cycle time, butmuch shorter than the averaging time for each mea-surement point ( ∼
20 seconds). Averaging the preces-sion about B N,z during time τ over a Gaussian distribu-tion of nuclear fields, gives a Gaussian coherence decay R ∞−∞ √ πσ e − B , z / σ cos( gµ b B N , z τ / ~ ) dB N,z = e − ( τ/T ∗ ) ,with T ∗ = √ ~ /gµ b σ ∼
30 ns [6, 7] (assuming σ =1.5mT, extracted from the Rabi oscillations, see [21]). Thisdecay is plotted in Fig. 2a (solid line). However, the ob-served Ramsey signal cannot be compared directly withthis curve because we have to take into account the ef-fect of the nuclear field during the π/ π/ B N , z shifts the electron spin reso-nance condition and as a result, the fixed-frequency RFpulses will be somewhat off-resonance which decreasesthe fidelity of the rotations.We include these effects in a simulation of the spindynamics, and consider from here on not just a singlespin but the actual two-spin system. At the end ofthe cycle, the two-spin state is given by ψ ( τ, B L , R ) = U L π ( B L ) U R π ( B R ) V L τ ( B L ) V R τ ( B R ) U L π ( B L ) U R π ( B R ) |↑↑i .Here, U L,Rθ ( B L , R ) is the single spin time-evolutionoperator (for an intented θ -rotation) resulting from thedriving field and the z -component of the nuclear fieldsin the left and right dot, B L and B R . The operator V L , R τ ( B L , R ) represents the single spin evolution duringa time τ in the presence of the nuclear field only. Wecan then compute P odd at the end of the pulse sequence,averaging over two independent Gaussian distributionsof nuclear fields in the left and right dot: P odd ( τ ) = 12 πσ Z Z e − ( B L + B R σ ) e P odd ( τ, B L,R ) dB L dB R ; e P odd ( τ, B L , R ) = |h ψ ( τ, B L , R ) | ↑↓i| + |h ψ ( τ, B L , R ) | ↓↑i| . This numerically calculated P odd ( τ ) is plotted in Fig. 2a(dotted line), after rescaling in order to convert P odd to acurrent flow I dot (see caption). We see that the predicteddecay time is longer when the rotations are imperfectdue to resonance offsets. This is more clearly visible inFig. 2b, where the computed curves are shown togetherwith Ramsey measurements for a wide range of drivingfields. The experimentally observed Ramsey decay timeis longer for smaller B ac , in good agreement with the nu-merical result. This effect can be understood by consid-ering that a burst doesn’t (much) rotate a spin when thenuclear field pushes the resonance condition outside theLorentzian lineshape of the excitation with width B ac .If the spin is not rotated into a superposition, it cannotdephase either. As a result, the cases when the nuclearfield is larger than the excitation linewidth do not con-tribute to the measured coherence decay. The recordeddephasing time is thus artificially extended when long,low-power RF bursts are used ( B ac / σ . τ + τ is shown in the main panel of Fig. 3a. We see im-mediately that the spin-echo decay time, T , echo , is muchlonger than the dephasing time, T ∗ .This is also clear from the data in Fig. 3c, which istaken in a similar fashion as the Ramsey data in Fig. 2c,but now with an echo pulse applied halfway through thedelay time. Whereas the fringes were fully suppressed fora 150 ns delay time without an echo pulse, they are stillclearly visible after 150 ns if an echo pulse is used. As a T ~(0.29±0.05) m s B =42 mT ext t
30 ns30 ns 60 ns t p /2 p p /2 a t + t (ns) T ~(33±22) ns initial -2 p -1 p p p Phase 3rd pulse D o t c u rr en t ( f A ) m m t - t (ns) -75 0 75 150 t + t =150 ns t + t =150 ns D o t c u rr en t ( f A ) b C FIG. 3: (color online) a) Spin-echo signal as a function of totalfree-evolution time τ + τ . Each data point represents thecurrent through the dots averaged over 20 seconds at constant B ext = 42 mT, f ac = 210 MHz, B ac = 3 mT. Dashed line:best fit of a Gaussian curve to the data in the range τ + τ =0 −
100 ns. Solid line: best fit of e − (( τ + τ ) /T , echo ) to the datain the range τ + τ = 100 −
800 ns. Dotted line: numericallycalculated dot current P odd ( m + 1)80 + 25 fA, taking σ = 1 . m = 1 .
83. Considerable scattering ofthe data points is not due to the noise of the measurementelectronics (noise floor about 5 fA), but caused by the slowevolution of the statistical nuclear field. Inset: spin-echo pulsesequence. b) Spin-echo signal as a function of τ − τ . Dashedline: best fit of a Gaussian curve to the data. c) Spin-echosignal for τ + τ = 150 ns as a function of the relative phasebetween the first two and third pulse. Dashed line is the bestfit of a cosine to the data. further check, we measured the echo signal as a functionof τ − τ (Fig. 3b). As expected, the echo is optimal for τ = τ and deteriorates as | τ − τ | is increased. The dipin the data at τ − τ = 0 has a half width of ∼
27 ns,similar to the observed T ∗ .Upon closer inspection, the spin-echo signal in Fig. 3areveals two types of decay. First, there is an initial decaywith a typical timescale of 33 ns (obtained from a Gaus-sian fit), which is comparable to the observed Ramseydecay time when using the same B ac . This fast initialdecay occurs because the echo pulse itself is also affectedby the nuclear field. As a result it fails to reverse theelectron spin time evolution for part of the nuclear spinconfigurations, in which case the fast dephasing still oc-curs, similar as in the Ramsey decay. To confirm this, wecalculate numerically the echo signal, including the effectof resonance offsets from the nuclear fields, similar as inthe simulations of the Ramsey experiment. We find rea-sonable agreement of the data with the numerical curve(dotted line in Fig. 3a), regarding both the decay timeand the amplitude.The slower decay in Fig. 3a corresponds to the lossof coherence that cannot be reversed by a perfect echopulse. We extract the spin-echo coherence time T , echo from a best fit of a + be − (( τ + τ ) /T , echo ) [11, 13] to thedata ( a, b, T , echo are fit parameters) and find T , echo =(290 ±
50) ns at B ext =42 mT (see Fig. 3a, solid line). Wenote that the precise functional form of the decay is hardto extract from the data, but fit functions of the form a + be − ( τ/T ) d with d between 2 and 4 give similar decaytimes.Measurements at higher B ext are shown in Fig. 4a,b.Here, experiments were only possible by decreasing thedriving field and as expected, we thus find a longer ini-tial decay time, similar as seen in Fig. 2b for Ramseymeasurements. The longer decay time from which we ex-tract T , echo tends to increase with field, up to 0.44 µ s at B ext =70 mT. This is roughly in line with the spin echodecay time of 1 . µ s observed for two-electron spin statesat B ext =100 mT [10].The field-dependent value for T , echo we find is morethan a factor of 10 longer than T ∗ , which is made possi-ble by the long correlation time of the nuclear spin bath.We now examine what mechanism limits T , echo . The z -component of the nuclear field can change due to the spin-conserving flip-flop terms H ff = ( S + h − + S − h + ) in thehyperfine Hamiltonian S · h , and due to the dipole-dipoleinteraction between neighbouring nuclear spins. Directelectron-nuclear flip-flop processes governed by H ff arenegligible at the magnetic fields used in this experiment,because of the energy mismatch between the electronand nuclear spin Zeeman splitting. However, the energy-conserving higher-order contributions from H ff can leadto flip-flop processes between two non-neighboring nu-clear spins mediated by virtual flip-flops with the electronspin [12, 13, 16, 22]. It is predicted that this hyperfine-mediated nuclear spin dynamics can lead to a field de-pendent free-evolution decay of about 0.1-100 µ s for thefield range 1-10 T [13, 14, 16]. Interestingly, some theo-retical studies [13, 22] have shown that this type of nu-clear dynamics is reversible (at sufficiently high field) byan echo-pulse applied to the electron spin. The coher-ence decay time due to the second possible decoherencesource, namely the dipole-dipole interaction, is expectedto be 10-100 µ s [11], independent of magnetic field (once B ext is larger than ∼ . t + t (ns) B =48 mT ext t p /2 p p /2 T ~(0.28±0.03) m s t T ~(40±18) ns initial T ~(0.44±0.05) m s T ~(53±27) ns initial t t B =70 mT ext t + t (ns) p /2 p p /2 D o t c u rr en t ( f A ) FIG. 4: (color online) a) Spin-echo signal at B ext = 48 mT( f ac = 280 MHz) and 70 mT ( f ac = 380 MHz). Pulse se-quence depicted in the insets. Solid and dashed lines are bestfits to the data as in Fig. 3a. long because (during the manipulation stage) the energyrequired for one of the electrons to be promoted to areservoir ( > µ eV) is much larger than the tunnel rate( < . µ eV). In principle, the Heisenberg coupling J be-tween the electron spins in the two quantum dots couldalso lead to decoherence, but during the manipulationstage, we expect that J is very small due to the largelevel detuning. Altogether, the most likely limitation tothe observed T , echo is hyperfine-mediated flip-flops be-tween any two nuclear spins.To conclude, we have performed time-resolved mea-surements of the dephasing of a single electron spin in aquantum dot caused by the interaction with a quasi-staticnuclear spin bath. We have largely reversed this dephas-ing by the application of a spin-echo technique. The echopulse extends the decay time of the electron spin coher-ence by more than a factor of ten. We obtain a T , echo of 0.29 µ s and 0.44 µ s at magnetic fields of 42 and 70mT respectively. While even longer coherence times areexpected at higher magnetic fields, the observed decaytimes are already sufficiently long for further explorationof electron spins in quantum dots as qubit systems.We thank D. Klauser, W. Coish, D. Loss, R. de Sousa,R. Hanson, S. Saikin, I. Vink and T. Meunier for dis-cussions; K.-J. Tielrooij for help with device fabrication;R. Schouten, A. van der Enden and R. Roeleveld fortechnical assistance and L. Kouwenhoven for mentorshipand support. We acknowledge financial support from theDutch Organization for Fundamental Research on Matter(FOM) and the Netherlands Organization for ScientificResearch (NWO). [1] D. Loss and D. P. DiVincenzo, Phys. Rev. A , 120(1998).[2] M. V. G. Dutt, et al. , Phys. Rev. Lett. , 227403 (2005).[3] A. S. Bracker, et al. , Phys. Rev. Lett. , 47402 (2005).[4] B. Herzog and E. L. Hahn, Phys. Rev. , 148 (1956).[5] J. P. Gordon and K. D. Bowers, Phys. Rev. Lett. , 368 (1958).[6] A. V. Khaetskii, D. Loss, and L. Glazman, Phys. 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