Qubit protection in nuclear-spin quantum dot memories
Z. Kurucz,1, 2 M. W. Sørensen,3 J. M. Taylor,4, 5 M. D. Lukin,5 and M. Fleischhauer1
1 Fachbereich Physik, University of Kaiserslautern, D-67663 Kaiserslautern, Germany Research Institute for Solid State Physics and Optics, H.A.S., H-1525 Budapest, Hungary 3 Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark 4 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 5 Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2

arXiv:0902.4566v1 [cond-mat.mes-hall] 26 Feb 2009

We present a mechanism to protect quantum information stored in an ensemble of nuclear spins in a semiconductor quantum dot. When the dot is charged the nuclei interact with the spin of the excess electron through the hyperﬁne coupling. If this coupling is made oﬀ-resonant it leads to an energy gap between the collective storage states and all other states. We show that the energy gap protects the quantum memory from local spin-ﬂip and spin-dephasing noise. Eﬀects of non-perfect initial spin polarization and inhomogeneous hyperﬁne coupling are discussed.

An essential ingredient for quantum computation and long-distance quantum communication is a reliable quantum memory. Nuclear spins in semiconductor nanostructures are excellent candidates for this task. With a magneton 3 orders of magnitude weaker than electron spins, they are largely decoupled from their environment, and the hyperﬁne interaction with electron spins allows one to access ensembles of nuclear spins in a controlled way [1, 2, 3, 4, 5, 6, 7, 8, 9]. In particular, the quantum state of an electron spin can be mapped onto the nuclear spins, giving rise to a long-term memory [3, 4, 5, 6]. Nevertheless, memory lifetimes are limited, e.g., by dipole-dipole interactions among the nuclei. In this Letter we demonstrate that the presence of the electron spin in the quantum dot substantially reduces the decoherence of this collective memory associated with surrounding nuclear spins. The virtual transitions between electronic and nuclear states can be used to produce an energy shift proportional to the number of excitations in the storage spin-wave mode. This isolates the storage states energetically and protects them against nuclear spin ﬂips and spin diﬀusion. Consider a quantum dot charged with a single excess ˆ electron as indicated in Fig. 1. The electron spin S is coupled to the ensemble of underlying nuclear spins ˆj I by the Fermi contact interaction,
N

e−

Energy

|0

∆K

|1q |1p , 1q

∆gap

|1

|1, 1q

FIG. 1: (Color online) Left: Charged quantum dot with a single, polarized excess electron. Right: Spectrum of the effective nuclear Hamiltonian in the presence of a polarized electron. Oﬀ-resonant hyperﬁne coupling results in a gap ∆gap between the storage state |1 and the non-storage states |1q . ∆K denotes the Zeeman shift due to the eﬀective magnetic ﬁeld associated with the electron spin (Knight shift).

of the nuclei [3, 4]. As will be shown here, the same can be used to provide a protective energy gap. Fully polarized nuclei. We start by reconsidering the storage of a qubit in a collective nuclear state [3]. In the simplest case when all the nuclear spins are initially polarized in the −z direction (zero temperature limit), the |↓ e and |↑ e spin states of the electron are mapped onto the nuclear spin states |0 ≡ |−I, −I, . . . , −I , |1 ≡
A ˆ Ω A+ |0

(2)

ˆ Hhf = A

ˆj ˆ ̺j Iz Sz +
j

1 2

ˆj ˆ ˆj ˆ I+ S− + I− S+

,

(1)

∝

j

̺j |−I, . . . , (−I + 1)j , . . . , −I , (3)

where A is the average hyperﬁne interaction constant, A ≈ 90 µeV for GaAs, and ̺j is proportional to the electron density at the position of the jth nucleus, j ̺j = 1. For convenience, we introduce the collective operators ˆ I A ≡ j ̺j ˆj . The ﬁrst term in Eq. (1) provides an eﬀecOH ˆ tive magnetic ﬁeld Bz = A Az /g ∗ µB for the electron, known as the Overhauser ﬁeld. The same also produces an energy shift for each nuclei, the so-called Knight shift. ˆ ˆ ˆ ˆ ˆ The ﬂip-ﬂop terms in Eq. (1), HJC = A (A+ S− + A− S+ ), 2 can be used to polarize the nuclear spins [1, 2], and to map the electron’s spin state into a collective spin mode

ˆ respectively. HJC couples the state |0 |↑ e to |1 |↓ e with 2 1/2 an angular frequency Ω = A . The detuning j ̺j 2I between these two states, δ = δ el + δ OH , comes from the electron’s intrinsic energy splitting δ el due to, e.g., an external magnetic ﬁeld, and from the Overhauser ﬁeld, δ OH = −AI. Coherent ﬂip-ﬂops between the electron and nuclear spins can be brought into resonance (δ ≪ Ω) through δ el , e.g., applying a spin-state dependent Stark laser pulse [10]. Then |0 (α|↓ e + β|↑ e ) can be rotated to (α|0 +β|1 )|↓ e , and the quantum information can be transferred from the electron to the nuclear spin ensemble

2 and back [3, 4]. Assume that, after the qubit has been written into the nuclei, the polarized electron is not removed from the dot but the hyperﬁne ﬂip-ﬂops are tuned oﬀ-resonant (δ ≫ Ω). Now real transitions can no longer take place between |1 |↓ e and |0 |↑ e . However, the residual virtual transitions repel the two states from each other, in analogy to the dynamic Stark eﬀect. As a result, after eliminating the electron, the energy of state |1 gets shifted by ∆gap = −Ω2 /4δ. The other, orthogonal states also having exactly one spin ﬂipped (denoted by |1q in ˆ Fig. 1) are “subradiant”, i.e., are not coupled via HJC to the electron. Therefore, they are unaﬀected by the shift. This is the origin of the energy gap. To understand the protection scheme, let us introduce nuclear spin waves. As long as the nuclei remain highly polarized, one can introduce bosonic operators through √ ˆj the Holstein-Primakoﬀ transformation: aj ≈ I− / 2I, ˆ √ ˆj ˆj ˆj ˆ a† ≈ I+ / 2I, and a† aj = Iz + I. This allows us to ˆj deﬁne the bosonic spin waves ˆ Φq ≡ ηqj aj , ˆ
j

ˆ Φ† ≡ q

∗ ηqj a† , ˆj j

(4)

where the unitary matrix ηqj describes the mode functions. We identify the storage mode q = 0 as the one √ ˆ given by η0j = 2I A ̺j , and write |1 = Φ† |0 . This is 0 Ω the mode which is directly coupled to the electron spin. ˆ ˆ ˆ ˆ ˆ In fact, HJC ≈ Ω Φ† S− + Φ0 S+ is a Jaynes-Cummings 0 2 coupling in the bosonic approximation. After eliminat2 ˆ ˆ ˆ ˆ ing the electron, HJC reduces to Hgap = − A A+ A− ≈ 4δ ˆ ˆ ˆ † Φ0 . As shown in Fig. 1, Hgap lifts the degeneracy ∆gap Φ0 between states of diﬀerent number of storage-mode excitations. This is the key feature of our protection scheme: any decoherence process that is associated with a tranˆ ˆ sition from the storage mode Φ0 to any other mode Φq now has to bridge an energy diﬀerence. If this gap is larger than the spectral width of the noise, the eﬀect of the noise is substantially reduced. A more detailed analysis shows that the oﬀ-resonant interaction with the electron spin—which itself is coupled, e.g., to phonons—leads in general also to an additional decoherence mechanism for the nuclear spins. If the corresponding electron spin dephasing rate γ is small compared to the electron’s precession frequency δ, the decay rate for the storage mode is reduced by the low probability of exciting the electron spin state: γΩ2 /δ 2 ≪ γ. In addition to the gap, the electron is also responˆ ˆ ˆ sible for the Knight shift HK = AAz Sz . The difference of the Knight shifts for the |0 and |1 states, 2 ∆K = − A j ̺3 j j ̺j , is typically much less than 2 ∆gap . When the hyperﬁne coupling is inhomogeneous, however, |1 fails to be eigenstate of the Knight shift 1 ˆ Hamiltonian: HK |1 = (− 2 δ OH + ∆K )|1 + ζ|1⊥ , where ⊥ the state |1 is orthonormal to |1 and the coupling pa2 2 2 4 rameter ζ 2 = A j ̺j − ∆K characterizes the j ̺j 4

inhomogeneities. As a consequence, the storage mode is only an approximate eigenmode, and it gradually mixes with non-storage modes as time passes. This causes loss of the stored qubit. |1⊥ is, however, oﬀ-resonant due to the energy gap, and our simulations show that the corresponding probability of ﬁnding the system in state 2 |1⊥ is bounded by 4ζ 2 /∆gap , so the detrimental eﬀect of the inhomogeneous Knight shift is suppressed by the energy gap. In addition, since the admixture of |1⊥ is a coherent process, it can be cancelled by refocusing (echo) methods. A large gap can be achieved by bringing the hyperﬁne interaction close to resonance. For example, a non-zero external magnetic ﬁeld or laser induced AC Stark shifts [10] can partially cancel the Overhauser ﬁeld, such that δ ≪ δel ≈ −δOH = AI. (Of course, δ should be kept sufﬁciently large so that the hyperﬁne coupling remains oﬀresonant). The requirement of separation of time scales implies ζ ≪ |∆gap | ≪ Ω ≪ |δ|, i.e., δ 10Ω. To estimate the orders of magnitude of the diﬀerent energies, we take an oblate Gaussian electron density of ratio (1, 1, 1/3), 1 and we consider spin- 2 nuclei. Then it is easy to see that ∆K and ζ are inversely proportional to the number of nuclei N , whereas Ω, ∆gap ∝ N −1/2 only (Fig. 2a). To analyze the decoherence suppression, we ﬁrst consider a simplistic noise model where the nuclear spins are coupled to ﬂuctuating, classical ﬁelds. The correspondˆ I ing interaction Hamiltonian is given by V = j Bj · ˆj . We assume isotropic Gaussian noise with zero mean and
j k Bµ (t)Bν (t′ ) = δµν ξjk Ce−Γ|t−t |
′

(5)

for µ, ν = x, y, z, where ξjk speciﬁes the spatial correlations of the noise acting on diﬀerent nuclei. For simplicLinear size of dot (σ) [in lattice units] 4 6 8 10 20 40 60 80 103 10 100 Ω 100 10 100 10 ∆gap 1 1 0.1 ∆K 0.1 0.01 10 0.01 10−3 ζ 10−3 10−4 ΓD 10−4 −5 10 1 10−5 10−6 10−6 10−7 ′ −7 ΓD (n = 0) 10 10−8 0.1 10−8 1 104 105 106 107
3

Energy scales [neV]

Ω ∆gap ∆K ζ ω
10 102 (N = 105 )

100 10 1 0.1

Number of nuclei (N ) (a)

Flipped spins (n) (b)

103

104

FIG. 2: Hyperﬁne Rabi frequency (Ω), protective energy gap (∆gap ), Knight shift diﬀerence between the logical states (∆K ), symmetry breaking couplings due to inhomogeneities (ζ and ω), qubit decoherence rate due to dipolar spin diﬀusion without (ΓD ) and with (Γ′ ) protection. (a) The fully D polarized (zero temperature) case is displayed as function of 1 the number of spin- 2 nuclei (N ) taking part in the storage, i.e., located within 3σ of the oblate Gaussian electron distribution with in-plane variance σ. (b) Estimated energies in dark states |Dn,β with n spins ﬂipped from the fully polarized state for N = 105 . Energy units are chosen to match GaAs.

Energy scales [MHz]

3
1

Ξ
0.1 10 10
−2

Ξ′

N = 105 104 N = 104 103 N = 103 102

N = 103 N = 104

−3

10

10−4

N = 105
10−2 0.1 1 10 102 103 104 10−2 0.1 1 10 102 103 104

1

Noise correlation lenght (ξ) [lattice units]

FIG. 3: The parameters Ξ and Ξ′ describing the eﬀects of spatial correlations in the classical noise (ξjk = e−rjk /ξ ) for diﬀerent number of nuclei. The same family of Gaussian electron densities was used as in Fig. 2. The bullets on the curves denote the linear size of the dot given by the variance σ.

ity, the noise spectrum is assumed to be Lorentzian with a width Γ, although similar results hold for other spectra with a high-frequency cut-oﬀ. j ˆj ˆ Let us ﬁrst discuss the dephasing part, Vz = j Bz Iz , of the noise. Using the bosonic spin-wave operators inˆ troduced in Eq. (4) we can express Vz as ˆ Vz =
j j Bz a† aj = ˆj ˆ pq j j ∗ ˆp ˆ Bz ηpj ηqj Φ† Φq .

(6)

Dephasing of individual nuclear spins thus means transfer of excitations between diﬀerent spin-wave modes. Especially, it leads to both real and virtual transitions from |1 to a non-storage state |1q (with q = 0). As the latter state is “subradiant” and, thus, equivalent to |0 when the memory is read out, this process essentially results in damping (for real transitions) and dephasing (for virtual transitions) of the stored logical qubit [11]. This can be seen by formally eliminating the classical ﬁelds and all non-storage mode in Markov approximation and deriving a master equation for the storage mode. For that, we assume the zero temperature limit with all non-storage ˆ modes Φq=0 in the vacuum state. This results in d ˆ ˆ ρ ρ = i ρ, Ez Φ† Φ0 + Lz (ˆ), ˆ ˆ 0 dt (7)

When the correlation length of the classical noise is smaller than the distance between the nuclei (local uncorrelated noise, ξjk ∼ δjk ), Ξ scales inversely with the number of nuclei (Fig. 3). In this case, the dephasing rate γ2 vanishes as 1/N , which is an eﬀect of the collective nature of the storage states [11]. The storage of a qubit corresponds to an encoding of the logical state in a large, delocalized ensemble of N physical spins. As the decoherence has strongly local character, there is only a very small eﬀect on the dephasing of the qubit. Secondly, the loss of the stored qubit is due to transitions among states with diﬀerent number of excitations in the storage mode. These transitions are strongly suppressed and the damping rate γ1 is decreased if ∆gap is large compared to the width of the noise spectrum Γ (or the corresponding cut-oﬀ frequency). Finally, we note that the opposite limit of inﬁnite spatial correlation length (ξjk = 1) corresponds to a homogeneous random ﬁeld resulting, e.g., from a global external source. In that case, Ξ ≈ 1 (see Fig. 3) and there is no protection against dephasing. Following a similar but slightly more involved proˆ cedure we can discuss the spin-ﬂip part Vxy = j ˆj j ˆj 1 j B+ I− + B− I+ of the noise. When deriving a 2 master equation for this case, we need to keep higher order terms in the Holstein-Primakoﬀ approximation: in √ ˆj the next order I− ≈ 2I 1 − λˆ† aj aj (and similarly aj ˆ ˆ ˆj ) with λ = 1 − (1 − 1/2I)1/2 . Here we have nefor I+ glected the probability of double or more excitations on the same site j, which is reasonable in the high polar1 ization (T = 0) limit and exact for spin- 2 nuclei. Omitting the energy shifts, the Lindbladian describing decoherences due to spin ﬂips reads, in leading order of 1/N , ˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆ Lxy (ˆ) = (γ3 + γ4 ) 2Φ0 ρΦ† − Φ† Φ0 ρ − ρΦ† Φ0 ρ 0 0 0 ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ + γ5 2Φ† Φ0 ρΦ† Φ0 − Φ† Φ0 Φ† Φ0 ρ − ρΦ† Φ0 Φ† Φ0 0 0 0 0 0 0 ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ + γ3 2Φ† ρΦ0 − Φ0 Φ† ρ − ρΦ0 Φ† , (10)
0 0 0

which describes decay with rate γ4 , dephasing with rate γ5 , and additionally thermalization (relaxation to the identity matrix) with rate γ3 . The rates read γ3 = Γ2 CΓIΞ′ , + (∆gap + ∆K )2 γ5 = γ4 = Γ2
j j

with energy shift Ez = (1 − Ξ)C∆gap /(Γ2 + ∆2 ) and gap ˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆ Lz (ˆ) = γ1 2Φ0 ρΦ† − Φ† Φ0 ρ − ρΦ† Φ0 ρ 0 0 0 ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ + γ2 2Φ† Φ0 ρΦ† Φ0 − Φ† Φ0 Φ† Φ0 ρ − ρΦ† Φ0 Φ† Φ0 . (8) 0 0 0 0 0 0 Here, γ1 is the damping rate of the stored qubit while γ2 describes its dephasing. The two rates are given by γ1 = CΓ (1 − Ξ), Γ2 + ∆2 gap γ2 = C Ξ, Γ (9)

4CΓIλ2 Γ2 + ∆2 K

̺4 j ̺2 j

2CΓIλ2 , + (∆gap − ∆K )2
2

.

(11)

where we have introduced the dimensionless parameter 2 2 containing the spatial part Ξ ≡ jk ξjk ̺2 ̺2 j k l ̺l of the noise correlator.

In the limit of vanishing spatial correlations of the spinﬂip noise, Ξ′ ≡ jk ξjk ̺j ̺k / l ̺2 tends to 1 (Fig. 3) l and we have protection against thermalization (γ3 ) because of the separation of |0 and |1 by ∆gap + ∆K . The decay corresponding to γ4 is due to spin-ﬂip induced transitions between |1 and |1p , 1q (the latter containing a total of two excitations but none in the storage mode), and the energy to bridge is in the order of ∆gap − ∆K (see Fig. 1). Finally, the last factor in the dephasing rate

4 γ5 scales as 1/N , indicating that it is the collective nature of the storage that leads to protection. Note that the nonlinearity of the Holstein-Primakoﬀ representation is responsible for this dephasing: the virtual non-storage excitations are interacting with the storage mode. Another potential source of decoherence is nuclear spin diﬀusion due to dipole-dipole interaction between nuclear spins [12]. The energy gap gives protection against this eﬀect, too. The dipolar interaction between the pairs of spins is described in the secular approximation by ˆ HD =
j=k

ˆj ˆk ˆj ˆk Bjk I+ I− − 2Iz Iz ≈ 2I

Bjk aj ak , (12) ˆ† ˆ
j=k

3 where Bjk = 1 γ 2 (3 cos2 θjk − 1)/rjk , γ is the gyromag4 netic factor, rjk = rj −rk is the distance between two nuclei, θjk is the zenith angle of the vector rjk , and we used the ﬁrst order Holstein-Primakoﬀ approximation. At full polarization, we can rewrite the dipolar Hamiltonian (12) in terms of the bosonic spin wave mode operators (4) ∗ ˜ ˆ† ˆ ˜ ˆ as HD = j=k Bjk ηpj ηqk . pq Bpq Φp Φq , with Bpq = Thus, the storage mode is coupled to a bath of nonstorage modes as if it were a central spin coupled to a mesoscopic spin bath [13, 14]. Although the total ˆ number of excitations is conserved, HD is responsible for decoherence of the qubit via transitions from the storage state |1 to non-storage states |1q . In fact, the non-storage modes produce a ﬂuctuating eﬀective transversal magnetic ﬁeld with (complex) Larmor fre˜ ˆ ˆ quency Ωeﬀ = 2 q=0 B0q Φq . If the electron were not D− present, these ﬂuctuations would lead to a decoherence ˜ rate ΓD ∼ ∆Ωeﬀ = 2 q=0 |B0q |2 1/2 in the fully poD larized state, which is numerically found to be in the order of 100 Hz for GaAs (Fig. 2a). With the protective gap, however, the storage mode creation and anniˆ ˆ hilation operators (Φ† and Φ0 ) rotate rapidly with re0 spect to the other ones, and the above coupling averages out and disappears in ﬁrst order of the dipolar perturbation. In second order, the strength of the remaining coupling between the storage mode and mode q ˜ ˜ is proportional to ∆−1 r=0 B0r Brq , and the corregap sponding ﬂuctuations yield a decoherence rate of Γ′ ∼ D ˜ ˜ 2 1/2 ∼ 3 × 104 Hz2 /∆gap . ∆−1 2 q=0 gap r=0 B0r Brq Typically, ∆gap ∼ 1 MHz depending on the dot size (Fig. 2a), so the eﬀects of spin diﬀusion can be suppressed by several orders of magnitude. Non-perfect spin polarization. Finally, we investigate the consequences of non-perfect nuclear spin polarization. It has been shown that partially polarized nuclei (at ﬁnite temperature) can also be used for storing a qubit state [4]. Instead of the fully polarized state (2), the initial preparation drives the nuclear ensemble into a statistical mixture of dark states |Dn,β deﬁned by ˆ A− |Dn,β = 0. These dark states can be characterized by the total number of spins ﬂipped n and the permutation group quantum number β. As the detuning δ is

adiabatically swept from far negative to far positive, a superposition of the |↓ e and |↑ e electron spin states is mapped into the mixture of superpositions of the nuclear A ˆ spin states |Dn,β and |En,β ≡ Ωn A+ |Dn,β , and the qubit state is eﬃciently written into the memory [4]. When the electron is left in the quantum dot, it feels diﬀerent Overhauser ﬁelds for diﬀerent dark states, hence OH the detuning should be adjusted such that δn + δ el ≫ OH Var(δn ). Moreover, the hyperﬁne Rabi frequency also varies with n and the energy gap ∆gap,n is not the same for all dark states. This inhomogeneous broadening would result in dephasing of the qubit, but can be avoided by a symmetric spin echo sequence [4]. To describe inhomogeneous eﬀects in the case of nonperfect polarization, ﬁrst we note that the storage state |Dn,β is no longer an eigenstate of the Knight shift operator, but it is partially mapped into an orthogonal state: OH ⊥ ˆ HK |Dn,β = − 1 δn |Dn,β + ωn |Dn,β . This is due to the 2 ˆ fact that the inhomogeneous Az,± operators do not follow the angular momentum commutation relation. Furtherˆ ˆ more, |En,β is neither an eigenstate of Hgap nor of HK : ˆ n,β = (− 1 δ OH + ∆K,n + ∆gap,n )|En,β + ζn |E ⊥ . H|E n,β 2 n The parameters can be expressed as expectation values in |Dn,β : ˆ ˆ Ω2 = A2 A− A+ , n
2 2 ˆz ˆ ωn = A A2 − Az 4 ˆ ˆ ˆ ˆ ∆gap,n = A4 A− A+ A− A+ 4δn Ω2 , n 3 ˆ ˆ ˆ A ˆ ∆K,n = 2 Az − A A− Az A+ 2Ω2 , n ˆ ˆ ζ 2 = En,β |H 2 |En,β − En,β |H|En,β 2 .

2

,

(13)

The explicit form of the inhomogeneous dark states [4] allows us to estimate these values (see Fig. 2b). We expect that the storage mode is still protected as long as ωn and ζn are much smaller than ∆gap , which is the case even for considerable unpolarized fraction (n/N ). In summary, we have demonstrated that it is possible to suppress the inﬂuence of spin-dephasing and spin-ﬂips on a quantum memory consisting of a delocalized ensemble of nuclear spins in a quantum dot if the noise has a highly local character and the spectral width or cutoﬀ frequency of the noise spectrum is small compared to the energy gap. We have shown in particular that the memory can be protected against nuclear spin diﬀusion mediated by dipole-dipole interaction. We have also analyzed the eﬀects of inhomogeneous hyperﬁne couplings and imperfect initial nuclear spin polarization. JMT wishes to thank the Fleischhauer group for their kind hospitality during his stay. This work was supported by the EU network EMALI; JMT is supported by Pappalardo.

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