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Effect of Exchange Interaction on Spin Dephasing in a Double Quantum Dot

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Effect of Exchange Interaction on Spin Dephasing in a Double Quantum Dot PHYSICAL REVIEW LETTERS week ending PRL 97, 056801 (2006) 4 AUGUST 2006 Effect of Exchange Interaction on Spin Dephasing in a Double Quantum Dot E. A. Laird,1 J. R. Petta,1 A. C. Johnson,1 C. M. Marcus,1 A. Yacoby,2 M. P. Hanson,3 and A. C. Gossard3 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel 3Materials Department, University of California at Santa Barbara, Santa Barbara, California 93106, USA (Received 3 December 2005; published 31 July 2006) We measure singlet-triplet dephasing in a two-electron double quantum dot in the presence of an exchange interaction which can be electrically tuned from much smaller to much larger than the hyperfine energy. Saturation of dephasing and damped oscillations of the spin correlator as a function of time are observed when the two interaction strengths are comparable. Both features of the data are compared with predictions from a quasistatic model of the hyperfine field. DOI: 10.1103/PhysRevLett.97.056801 PACS numbers: 73.21.La, 71.70.Gm, 71.70.Jp Implementing quantum information processing in solid- Enuc. When J Enuc, we find that PS decays on a time @ state circuitry is an enticing experimental goal, offering the scale T2 =Enuc 14 ns. In the opposite limit where possibility of tunable device parameters and straightfor- exchange dominates, J Enuc, we find that singlet corre- ward scaling. However, realization will require control lations are substantially preserved over hundreds of nano- over the strong environmental decoherence typical of seconds. In the intermediate regime, where J Enuc,we solid-state systems. An attractive candidate system uses observe oscillations in PS that depend on the ratio Enuc=J. electron spin as the holder of quantum information [1,2]. In Our results show that a finite exchange energy can be used III-V semiconductor quantum dots, where the highest de- to extend spin correlations for times well beyond T2 . gree of spin control has been achieved [3–9], the dominant These observations are in reasonable agreement with decoherence mechanism is hyperfine interaction with the recent theory, which predicts a singlet probability (assum- 0 lattice nuclei [10]. A recent experiment [9] studied this ing perfect readout) PS S that exhibits damped oscilla- decoherence in a qubit encoded in a pair of spins [11]. In tions as a function of time and a long-time saturation that this situation, the dynamics are governed by two compet- depends solely on the ratio Enuc=J [16]. To compare ex- ing effects: the hyperfine interaction, which tends to mix periment and theory quantitatively we introduce an empiri- the singlet and triplet basis states, and exchange, which cal visibility, V, to account for readout inefficiency, 0 tends to preserve them. PS S1 ÿ V1 ÿ PS S. The interplay of hyperfine and exchange effects has been The device used in the experiment, shown in Fig. 1(a),is studied recently via spin-blockaded transport in two fabricated on a GaAs=Al0:3Ga0:7As heterostructure with a double-dot systems [12,13]. Oscillations and bistability two-dimensional electron gas (density 2 1015 mÿ2, mo- [12] of the leakage current, as well as suppression of bility 20 m2=Vs) 100 nm below the surface. Ti=Au top mixing with stronger exchange [13] were observed. The gates define a few-electron double quantum dot. The in- topic also has a long history in physical chemistry: recom- terdot tunnel coupling tc and 0; 2- 1; 1 detuning are bination of a radical pair created in a triplet state proceeds also separately tunable. A charge-sensing quantum point 2 significantly faster for radicals containing isotopes whose contact with conductance gs 0:2e =h allows the occu- nuclei carry spin [14]. By lifting the singlet-triplet degen- pancy of each dot to be separately measured [17,18]. We eracy, the exchange interaction suppresses spin transitions; monitor gs using a lock-in amplifier with a 1 nA current its strength can be deduced from the magnetic field depen- bias at 335 Hz, with a 30 ms time constant. dence of the recombination rate [15]. However, exchange Measurements were made in a dilution refrigerator at is difficult to tune in situ in chemical systems. electron temperature Te 100 mK measured from the In this Letter, singlet correlations between two separated width of the 1; 1- 0; 2 transition [19]. Gates L and R electrons in a GaAs double-dot system are measured as a (see Fig. 1) were connected via filtered coaxial lines to the function of a gate-voltage tunable exchange J and as a outputs of a Tektronix AWG520. We report measurements function of time S following the preparation of an initial for two settings of tunneling strength, controlled using singlet. This study gives insight into the interplay of local voltages on gate T and measured from the width of the hyperfine interactions and exchange in a highly control- 1; 1- 0; 2 transition: tc 23 eV (‘‘large tc’’) and tc < lable quantum system. We measure the probability PS S 9 eV (‘‘small tc’’) [19]. Except where stated, measure- that an initial singlet will be detected as a singlet after time ments were made in a perpendicular magnetic field of S for J ranging from much smaller than to much greater 200 mT, corresponding to a Zeeman energy EZ than the rms hyperfine interaction strength in each dot, 5 eV Enuc. 0031-9007=06=97(5)=056801(4) 056801-1 © 2006 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 97, 056801 (2006) 4 AUGUST 2006 a)T b) T- E pulse cycle is spent at M, the relatively slow measurement T Z 0 of the sensor gs gives a time-averaged charge configuration T g + at the M point. This signal is calibrated to give a singlet L R s 2tc state probability PS S by comparing values within the c) 0 15 J pulse triangle with values within 1; 1 (which defines δg (10-3e2/h) S -455 PS 0) and within 0; 2 outside the pulse triangle (which (0,2)S S (1,2) S defines PS 1). P We first measure J , Enuc, and V at two values of tc, -458 (1,1) P' time (mV) P ,M τ allowing the saturation probability P 1 to be measured L S S S V (0,2) as a function of J. This saturation probability is found to -461 (0,1) M P depend on the ratio Enuc=J approximately as predicted by -335 -332 -329 0 ε theory [16]. We then measure the time evolution PS S, V (mV) R which shows damped oscillations, also in reasonable agreement with theory [16]. J is measured using the FIG. 1 (color). (a) Micrograph of a device with the same gate Rabi (or Larmor) sequence described in Ref. [9], in which design as the one measured (scale bar 500 nm). Voltages an adiabatic (compared with E ) ramp over 1 s to 1; 1 applied to gates L and R adjust the double-dot detuning, . nuc is used to prepare and measure the electron spin state in the Gate T sets the interdot tunnel coupling. The conductance gs of a nearby sensor quantum point contact monitors the average fj "#i; j #"ig basis. An exchange pulse produces coherent occupation of each dot. (b) Upper panel: Level diagram for the rotations with a period tR [shown in Fig. 2(a)] from which double dot near the 1; 1- 0; 2 transition ( 0) plotted versus we deduce the exchange coupling J h=tR [22]. Exchange (J) and Zeeman (EZ) energies are indicated. The Values of J for small and large tc are shown in symbol ᭹ denotes the S-T degeneracy. Labels m; n denote the Fig. 2(b), along with a fit to an empirical power-law form occupancies of the left and right dot, respectively. Lower panel: J / ÿ , giving 1:4 [23]. In Fig. 2(c), these values of The pulse scheme, consisting of prepare (P, P0), separate (S), J are compared with the results of an alternative method and measure (M) steps. Approximately 90% of the cycle is spent in which rapid dephasing at the S-T degeneracy produces in M. (c) gs close to the 1; 1- 0; 2 transition during application a dip in PS when the value of at the S point satisfies of pulses, showing the pulse triangle (marked) and the positions J E . J can then be measured from a knowledge of points P, P0, S, and M. A background plane has been Z subtracted. of the field, using EZ gBB, where B is the Bohr magneton, and taking the value g ÿ0:44, measured (using an in-plane field) in a different quantum dot device Figure 1(b) shows the relevant energy levels near the 1; 1- 0; 2 charge transition as a function of energy detun- ing between these charge states. With tc 0, the 1; 1 a) c) 0.8 1.0 P ( =200 ns) 200 small t S S singlet S and m 0 triplet T are degenerate; the m c s 0 s large t 1 triplets T are split off in energy from T by E . c 0 Z 0.6 (ns) R Finite tc leads to hybridization of the 0; 2 and 1; 1 t 100 singlets, inducing an exchange splitting J between S and 20 T . The 0; 2 triplet (not shown) is split off by the much 0 0 larger intradot exchange energy J 0;2 600 eV [20] and -2.0 -1.5 -1.0 -0.5 0.4 (mV) J( eV) is inaccessible.
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