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Nuclear spin polarization in quantum dots – The homogeneous limit
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Solid State Sciences 11 (2009) 965e969 www.elsevier.com/locate/ssscie
Nuclear spin polarization in quantum dots e The homogeneous limit H. Christ*, J.I. Cirac, G. Giedke MaxePlanckeInstitut fu¨r Quantenoptik, HanseKopfermanneStrasse, 1, D-85748 Garching, Germany Received 6 June 2007; received in revised form 12 September 2007; accepted 28 September 2007 Available online 9 October 2007
Abstract We theoretically discuss the dynamical quantum process of polarizing the nuclear spins in a quantum dot via the hyperfine interaction with the electron spin. The limit of homogeneous couplings, i.e. a flat electronic wave function, is analyzed in detail and approximate analytical solutions are shown to provide accurate results, allowing for the determination of cooling potential and rates, both of which reflect the effect of polarization-limiting dark states. We further provide a detailed microscopic description of these states that proves useful for the analysis of the effect of electron wave function changes during the cooling procedure. 2007 Elsevier Masson SAS. All rights reserved. PACS numbers: 71.70.Jp; 73.21.La Keywords: Nuclear spins; Quantum dots; Quantum information
1. Introduction Nuclear spin polarization in quantum dots (QD) has received tremendous attention in recent years (see Ref. [1] and references therein), because of its strong influence on the electron spin coherence, which is crucial for quantum computation [2], and because the nuclear spins themselves are an interesting physical system with possibly long coherence times [3] and interesting many body (dipolar) effects [4,5]. While most previous studies neglected quantum effects in the cooling process it has been shown [1,6] that coherences between nuclei can indeed strongly influence and limit the polarization process. We present here further evidence for these observations by considering the limit of a spatially homogeneous electron wave function, which is commonly assumed in studies of electron spin decoherence. Indeed we show in the present work that for the cooling process the level of homogeneity plays an important role. Naturally our theory is directly applicable to QD potentials and shapes producing a quasi-homogeneous electron
* Corresponding author. Tel.: þ49 089 32905368. E-mail address: [email protected] (H. Christ). 1293-2558/$ - see front matter 2007 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2007.09.027
distribution. Furthermore, as an inhomogeneous wave function can be approximated by a step function and our results bear relevance for the dynamics in each of the steps, the results presented here can be transferred to more general situations (cf. e.g. Ref. [7]). We consider the cooling of N nuclear spins in a QD through interaction with polarized electrons, as in Ref. [1]. One cooling cycle consists of (i) initialization of the electron spin in a well-defined direction j[i, and (ii) evolution of the combined system for a ‘‘short’’ time Dt. The hyperfine (HF) interaction for homogeneous coupling constants reads H¼
A Aþ I S þ Sþ I þ I z Sz þ Bext Sz : 2N N
ð1Þ
S is the spin operator for Pthe electron, A the HF coupling constant, Z ¼ 1, and I m ¼ i Iim are the three components of the homogeneous collective nuclear spin operators (m ¼ , z). The collective operators fulfill the angular momentum commutation relations [Iþ, Iz] ¼ Iþ, [Iþ, I] ¼ 2Iz. We take I ¼ 1/2 and neglect nuclear dipolar interactions, nuclear Zeeman energies and species inhomogeneities (see Ref. [1] for
H. Christ et al. / Solid State Sciences 11 (2009) 965e969
d A z A2 Dt z z r¼i I ; I ;r I ;r dt 2N 8N 2 2 A Dt þ I I r þ rI þ I 2I r I þ : 2 8N
ð2Þ
Although we consider in our model the polarization process in a single dot, the results bear relevance for spin cooling in the recently investigated double QDs [8,9]. There, the polarizing dynamics happens in the subspace of a two-electron singlet and a polarized triplet; coupling between the two is mediated by the difference dI ¼ ðIL IR Þ=2 of the collective nuclear spin operators of the two dots L and R, while the effective Overhauser field is given by the sum ðILz þ IRz Þ=2. In the homogeneous limit these operators obey the same algebra as the ‘‘bare’’ ones of our model, thus making our results directly applicable. 2. Achievable polarization Due to the collective coupling, singlet-like ‘‘dark states’’ can trap spin excitation and prevent cooling to the ground state. This effect is for homogeneous couplings most conveniently described in the Dicke basis [10], which is wellknown for example from the literature on superradiance. The basis states are jI; m; bi where I(I þ 1) is the eigenvalue of the collective angular momentum operator I2, and the eigenvalue of Iz is given by m. The possible values of I are N/2, N/2 1, ., the smallest value being 0 if N is even and 1/2 if N is odd; jmj I. The b is the permutation quantum number, which labels the different degenerate states with the same quantum numbers I and m. The degeneracy D of the states depends only on I and not on m, N N DðIÞ ¼ ð N=21 Þð N=2I1 Þ. When acting on a state jI; m; bi, the operator I decreases the quantum number m by one I jI; m; bi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi IðI þ 1Þ mðm 1ÞjI; m 1; bi. For m ¼ I the action of I yields zero, and thus the states jI; I; bi can not be polarized further in a system evolving according to Eq. (2). The steady state of the cooling process described by the master Eq. (2) starting from a completely mixed state is thus X DðIÞð2I þ 1ÞjI; IihI; Ij; ð3Þ rss ¼ 1=2N I
where the trace over the permutation quantum number b has been performed. We now evaluate the polarization of the steady state Eq. (3) normalized to the polarization of the ground state " # 2 4ðN 1Þ! 1 N hI z iss N ¼ 2 ; ðN=2Þ hI z i0 2N N ð½ðN 1Þ=2!Þ2 2
where for the fully polarized state one has a polarization hI z i ¼ N=2. As an example this expression can applied to the case N ¼ 2, where one finds a final polarization of 3/4. This is the expected result because one out of the four two-spin states cannot be cooled (the singlet). In Fig. 1 it is shown that for increasing particle number the possible polarization quickly decreases from 75% of the two-particle case. The negative indications for cooling purposes are obvious. pffiffiffiffiffiffi Application of the Stirling formula N!z 2pN Nþ1=2 eN gives the approximate result for N[1 hI z iss ¼ hI z i0
rffiffiffiffiffiffiffi 8 þ Oð1=NÞ: pN
ð4Þ
For the mesoscopic particle numbers of interest in the study of quantum dots, the obtainable polarization is thus negligible. In Fig. 1 it is shown that the approximate first order formula matches the exact value very well, as expected. 3. Time evolution The study of the time evolution of the nuclear spin system under the master Eq. (2) is similar to the study of the evolution of the atomic population in superradiant light emission from an ensemble of atoms. The latter problem has received considerable attention in the quantum optics literature, see e.g. the textbook [11] and review [12]. The focus of these studies was slightly different than it is in the cooling protocol we are considering. In quantum optics an ensemble with all atoms excited is the most studied situation, because for this initial condition one has pronounced signatures in the emitted light pulses and it is experimentally accessible. Here on the other hand, we are dealing with a completely mixed initial state. This complicates the situation because in this mixture superradiant as well as subradiant states occur. Nevertheless we 0.055
x 10-3
0
0.05 -0.2
Difference
a discussion of these effects). If the relevant system dynamics happens on a timescale larger than Dt, as is the case in the present study, we arrive at the continuous version of the master equation of Ref. [1] for the density matrix r of the nuclei
0.045
Coolable Fraction
966
0.04
-0.4 -0.6 -0.8 -1
0.035
-1.2
0.03
0
2000
4000
6000
8000 10000
N
0.025 Exact
0.02
Stirling 0.015
0
2000
4000
6000
8000
10000
N Fig. 1. The fraction of the nuclei that can be polarized in the case of homogeneous coupling constants. Shown are both the exact and the approximate numbers; the inset shows the difference between the two.
H. Christ et al. / Solid State Sciences 11 (2009) 965e969
where we have introduced k ¼ A2 Dt=8N 2 . The equation for the polarization hI z i does not close, but couples to the variable hI þ I i. The time evolution of this quantity depends on hI þ I z I i. Continuing this procedure leads to a hierarchy of coupled equations. Following the quantum optics literature on superradiance we make a factorization assumption z 2
2 ð5Þ ðI Þ ¼ hIz i ; hIiz Ijz i
hIiz ihIjz i.
which is equivalent to the assumption that ¼ In the following we apply this factorization to obtain analytical results for the time evolution of the polarization and check its validity by comparison with the exact numerical solution. Using this assumption the equation of motion for the polarization becomes
d z 2 hI i ¼ 2k I2 hI z i þhI z i ; dt
ð6Þ
where I2 ¼ ðI z Þ2 þ ð1=2ÞðI þ I þ I I þ Þ was used. For each given initial state jI; m; bi this equation is solved by hI z ðtÞiI;m ¼ 1=2 nI tan hðnI ð2kt c0 ÞÞ;
ð7Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where nI ¼ 4IðI þ 1Þ þ 1=2 and the initial condition depends on the value of m and enters through the constant c0 ¼ arctanh½2ðm 1=2Þ=nI =ð2nI Þ. In Fig. 2 we compare the evolution of the polarization averaged over all initial states X DðIÞhI z ðtÞiI;m ð8Þ hI z ðtÞiaveraged ¼ I;m
with the exact numerical calculation. The curves show good agreement, which serves as an indication that the factorization assumption captures the correct physics of the polarization process. To gain further insight into the evolution of the population, we consider the evolution of a pure initial state with total angular momentum and z-projection equal to the mean value of thePtotally mixed state, i.e. m ¼ 0 and Imean ¼ N=2 ð1=2N Þ I¼0 Ið2I þ 1ÞDðIÞ. Using the same approximations as for the evaluation of the total obtainable polarizations pffiffiffiffiffiffiffiffiffiffiffiffi (Eq. (4)) one gets Imean ¼ 2N=p þ Oð1Þ. In this limit the polarization develops according to 1 hI z ðtÞimean ¼ 2
! rffiffiffiffiffiffi rffiffiffiffiffiffi 8N 8N A2 Dt t ; tan h p p 8N 2
ð9Þ
0.2
0.15
/0
extract and present here approximate explicit analytic expressions for the evolution of the nuclear spin polarization. Instead of solving the dynamics of the density matrix of the nuclei, we concentrate on the evolution of the variable of our interest, hI z i. It obeys the equation of motion
d z zd hI i ¼ tr I r ¼ 2k I þ I ; dt dt
967
0.1
Exact 0.05
Mean Approx. Mean Averaged
0
0
1
2
t/A
3
4
x 104
Fig. 2. Comparison between the various (semi-)analytical solutions based on Eq. (5) and the exact numerical result for N ¼ 100 particles. The ‘‘average’’ (dashed) line is averaged over all states in the initially totally mixed nuclear state, see Eq. (8). The two other curves are directly obtained from Eq. (7) by substituting one specific value for the angular momentum I that enters the solution as a parameter; for the dashededotted line the average value of the total angular momentum I has beenpcalculated exactly and for the dotted ffiffiffiffiffiffiffiffiffiffiffiffi curve the approximate expression I ¼ 2N=p was used.
pffiffiffiffi where we used N [1. Fig. 2 shows the evolution in the ‘‘mean state’’ for both the exact and the approximate expressions for the mean value of the average spin Imean. Using the exact mean the results are very close to the actual evolution: Both the timescale and the final polarization are correctly predicted. When the approximate expression for the mean is employed, the timescale of the process is still correct. However, the predicted final polarization differs from the real polarization. This is to be expected, since the approximated mean value of I is only correct to order 1, which translates into a Oð1=NÞ error for the polarization. From Eq. (9) we can thus extract the timescale for reaching the steady state of the polarization pffiffiffiffiffiffi N 8p ; thom ¼ A 0:1
ð10Þ
pffiffiffiffi where we used Dt ¼ 0:1 N =A, i.e. Dt is chosen such that the condition necessary for the expansion of the time evolution operator is fulfilled. 4. Microscopic description of dark states In this section a new microscopic description of dark states is introduced, which, besides providing handy intuition, will serve in Section 5 as the basis for our estimations regarding ground state cooling in homogeneous situations. We prove that one can understand all dark states I jI; I; bi ¼ 0 as a superposition of states describing n ¼ N/2 I singlets and 2I ¼ N 2n polarized spins jYi. Let us denote a singlet of spins i and j as jSij i and by jS si 2S i¼1 ; Yi the state with S singlets jSsi siþ1 i and the remaining states spins in state jYi. The
H. Christ et al. / Solid State Sciences 11 (2009) 965e969
968
idea is to show that the statistical mixture of all permutations of jS i 2S i¼1 ; Yi PI ¼
1 X y XðpÞPjSf1;.;2Sg;Yi XðpÞ N! p˛SN
(where for a permutation p ˛ SN let XðpÞjSfsi g; Yi ¼ jSfpðsi Þg; Yi) is equal to the projector on the the (I, I )subspace:
ith Hadamard mode could be achieved, e.g. by performing phase flips jYij / jYij for all j with (Hm)ji ¼ 1. When applied after cooling with the first mode, such a mode change has no effect on nuclei already in state jYi but singlets are transformed to triplets (with zero z-projection) if the phase of only one of the two spins is flipped, so that the state in the new mode is X
pðnÞ
n
To see this, we first observe that SN Hp 1XðI;MÞ ðpÞ˛M2N with X(I, M )(p) acting (as above) on the space spanned by the vectors jI; M; bi with b ¼ 1; .; DðIÞ a label for the eigen ! values of a set of permutation operators extending I 2 ; and Iz to a complete commuting set (e.g. P2i1;2i ; i ¼ 1; .; N=2 cf. [10]) is a representation of the permutation group SN. The representation must be irreducible, since if it were not, there would exist at least two sets of vectors in HðI;MÞ that are not connected by SN, hence the projector on one of those would ! be an operator commuting with f I 2 ; I z ; P2i1;2i g, contradicting that this is a complete set of commuting observables on ðC2 Þ5N . Finally, we clearly have XI,I(p)PIX(I,I )(p)y ¼ PI and by Schur’s Lemma (which states that any operator commuting with all elements of a irrep is proportional to the identity see, e.g. Ref. [13]) it follows that PI f1I;I . Counting the number of permutations respectively the degeneracy of the (I, I )-subspace gives the normalization factor. This shows that the projector on the (I, m)-subspace is given by the equal-weight mixture of all permutations of N/2 I singlets and the remaining 2I spins in the state jI; mi. 5. Mode changes To cool beyond the dark state, the coupling operator I needs to be changed. In view of the analogy between a (collectively addressed) system of spins and a bosonic mode, addressing a differsuch a transformation of I can be viewed asP ent ‘‘mode’’ of the system [1]. Any ~I ¼ i gi Ii with ! g not parallel to ð1; .; 1Þ represents a different mode in this terminology. The number of modes that are needed to cool the nuclear spins to the ground state is in general not easily estimated due to the fact that orthogonal spin modes do not commute [14]. In order to get an estimate for the number of modes needed for cooling the spins, we use the simple Hadamard basis for the modes. The Hadamard matrices are recursively defined by H0 ¼ 1, and 1 Hm Hm : Hmþ1 ¼ pffiffiffi 2 Hm Hm The size of the matrix is 2m. Its columns are mutually orthogonal and represent the different modes we aim to address. Changing from the standard homogeneous coupling to the
qðn;uÞXðpÞPjS1;.;2ðnuÞ;Yi XðpÞy 5jT0 ihT0 j5u
p˛SN u¼0
where q(n, u) denotes the probability that u of the n singlets become a triplet after the mode change. We assume that changing from one Hadamard mode to the next typically breaks up half the singlets in each of the terms in the mixture, i.e. u ¼ n=2. We now calculate the total angular momentum I2 of this state, by first noting that in each of the above terms in the mixture one can split the collective operators up into terms that act on spins which remained untransformed by the mode change, and the rest Sall spins ¼ ðSin some T0 þ Srest Þ. The mixed terms between the two sums can be seen to be 0 readily. The terms involving the unchanged (rest) spins have the same total spin as before the mode change: Only singlets have been taken away. For the spins that have turned from singlets to triplets one can easily verify that their contribution to the new total angular momentum is (N/2 Iold)/4. In a semiclassical treatment we now calculate in the way outlined above the average angular momentum after k mode changes and arrive at the recursive formula Ikþ1 ðIkþ1 þ 1Þ ¼ Ik ðIk þ 1Þ þ ðN=2 Ik Þ=4;
ð11Þ
where k is the index for the number of modes changes, and from which we extract the minimal compatible polarization, Izk ¼ Ik. The numerical solution of the above recursive formula presented in Fig. 3 indicates that order of OðNÞ modes
0.8 0.7 0.6
/0
1 X
I; I; b I; I; b : PI ¼ DðIÞ b
n XX
0.5 0.4 0.3 0.2 0.1 0
0
500
1000
1500
2000
# modes k Fig. 3. Estimate of the achieved cooling after a given number of (Hadamard-) mode changes for N ¼ 103 nuclear spins.
H. Christ et al. / Solid State Sciences 11 (2009) 965e969
are necessary to completely cool the nuclear spins. We have verified this scaling behavior for a wide range of particle numbers. Note that a scheme employing Hadamard modes is not optimal: if one had complete freedom in the choice of modes, one could cool each spin individually and therefore cool everything to the perfect ground state in N steps; which is already better than the cooling procedure shown in Fig. 3. We are thus lead to the conclusion that achieving high polarization for strictly homogeneous couplings is hard: Sophisticated and undesirable control mechanisms for the electron wave function might be needed. As seen in Ref. [1], an inhomogeneous coupling will significantly simplify the cooling process.
6. Conclusions We have presented a study of nuclear spin cooling in quantum dots in the limit of a homogeneous electron wave function. We calculated exactly the cooling potential, presented analytical formulas for the time evolution of the polarization and, after introducing a handy intuition for the microscopic description of the dark states in the nuclear ensemble, showed that many different (Hadamard-) electron wave function profiles are needed to completely polarize the nuclei. The results underline the importance of the inhomogeneous Knight shift in nuclear spin cooling, which mitigates these limitations so strongly, that polarizations above 90% are achievable in realistic setups [1].
969
References [1] H. Christ, J.I. Cirac, G. Giedke, Phys. Rev. B 75 (2007) 155324. [2] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120 cond-mat/ 9701055. [3] J.M. Taylor, G. Giedke, H. Christ, B. Paredes, J.I. Cirac, P. Zoller, M.D. Lukin, A. Imamo˘glu, 2004, cond-mat/0407640. [4] D.H. Feng, I.A. Akimov, F. Henneberger, Phys. Rev. Lett. 99 (2007) 036604 cond-mat/0703386. [5] P. Maletinsky, A. Badolato, A. Imamoglu, Phys. Rev. Lett. 99 (2007) 056804. [6] A. Imamo˘glu, E. Knill, L. Tian, P. Zoller, Phys. Rev. Lett. 91 (2003) 017402 cond-mat/0303575. [7] S.I. Erlingsson, Y.V. Nazarov, Phys. Rev. B 70 (2004) 205327. [8] J. Baugh, Y. Kitamura, K. Ono, S. Tarucha, Phys. Rev. Lett. 99 (2007) 096804. [9] J.R. Petta, J.M. Taylor, A.C. Johnson, A. Yacoby, M.D. Lukin, C.M. Marcus, M.P. Hanson, A.C. Gossard, 2007, arXiv:0709.0920. [10] F.T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, Phys. Rev. A 6 (1972) 2211. [11] A.V. Andreev, V.I. Emelyanov, Y.A. Ilinski, Cooperative Effects in Optics: Superradiance and Phase Transitions, IOP Publishing, London, 1993. [12] M. Gross, S. Haroche, Phys. Rep. 93 (1982) 301. [13] T. Inui, Y. Tanabe, Y. Onodera, Group Theory and its Applications in Physics, second ed, Springer, Berlin, 1996. [14] In the bosonic approximation [1] the number of required modes is easily seen to be N, because the modes are mutually independent and initially all N modes are occupied and consequently have to be cooled one by one (neglecting the effects of inhomogeneous Knight fields). The bosonic model is only exact for high polarization, and in general is expected to be a lower bound on the performance of the cooling protocol for the spins. It is unclear, if the scaling behavior of the number of required modes is the same pffiffiffiffifor spins. One might think, for example, that one needs pffiffiffiffi only Oð N Þ modes in the spin case, because in the first mode w N excitations can be removed from the spin system.
Solid State Sciences 11 (2009) 965e969 www.elsevier.com/locate/ssscie
Nuclear spin polarization in quantum dots e The homogeneous limit H. Christ*, J.I. Cirac, G. Giedke MaxePlanckeInstitut fu¨r Quantenoptik, HanseKopfermanneStrasse, 1, D-85748 Garching, Germany Received 6 June 2007; received in revised form 12 September 2007; accepted 28 September 2007 Available online 9 October 2007
Abstract We theoretically discuss the dynamical quantum process of polarizing the nuclear spins in a quantum dot via the hyperfine interaction with the electron spin. The limit of homogeneous couplings, i.e. a flat electronic wave function, is analyzed in detail and approximate analytical solutions are shown to provide accurate results, allowing for the determination of cooling potential and rates, both of which reflect the effect of polarization-limiting dark states. We further provide a detailed microscopic description of these states that proves useful for the analysis of the effect of electron wave function changes during the cooling procedure. 2007 Elsevier Masson SAS. All rights reserved. PACS numbers: 71.70.Jp; 73.21.La Keywords: Nuclear spins; Quantum dots; Quantum information
1. Introduction Nuclear spin polarization in quantum dots (QD) has received tremendous attention in recent years (see Ref. [1] and references therein), because of its strong influence on the electron spin coherence, which is crucial for quantum computation [2], and because the nuclear spins themselves are an interesting physical system with possibly long coherence times [3] and interesting many body (dipolar) effects [4,5]. While most previous studies neglected quantum effects in the cooling process it has been shown [1,6] that coherences between nuclei can indeed strongly influence and limit the polarization process. We present here further evidence for these observations by considering the limit of a spatially homogeneous electron wave function, which is commonly assumed in studies of electron spin decoherence. Indeed we show in the present work that for the cooling process the level of homogeneity plays an important role. Naturally our theory is directly applicable to QD potentials and shapes producing a quasi-homogeneous electron
* Corresponding author. Tel.: þ49 089 32905368. E-mail address: [email protected] (H. Christ). 1293-2558/$ - see front matter 2007 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2007.09.027
distribution. Furthermore, as an inhomogeneous wave function can be approximated by a step function and our results bear relevance for the dynamics in each of the steps, the results presented here can be transferred to more general situations (cf. e.g. Ref. [7]). We consider the cooling of N nuclear spins in a QD through interaction with polarized electrons, as in Ref. [1]. One cooling cycle consists of (i) initialization of the electron spin in a well-defined direction j[i, and (ii) evolution of the combined system for a ‘‘short’’ time Dt. The hyperfine (HF) interaction for homogeneous coupling constants reads H¼
A Aþ I S þ Sþ I þ I z Sz þ Bext Sz : 2N N
ð1Þ
S is the spin operator for Pthe electron, A the HF coupling constant, Z ¼ 1, and I m ¼ i Iim are the three components of the homogeneous collective nuclear spin operators (m ¼ , z). The collective operators fulfill the angular momentum commutation relations [Iþ, Iz] ¼ Iþ, [Iþ, I] ¼ 2Iz. We take I ¼ 1/2 and neglect nuclear dipolar interactions, nuclear Zeeman energies and species inhomogeneities (see Ref. [1] for
H. Christ et al. / Solid State Sciences 11 (2009) 965e969
d A z A2 Dt z z r¼i I ; I ;r I ;r dt 2N 8N 2 2 A Dt þ I I r þ rI þ I 2I r I þ : 2 8N
ð2Þ
Although we consider in our model the polarization process in a single dot, the results bear relevance for spin cooling in the recently investigated double QDs [8,9]. There, the polarizing dynamics happens in the subspace of a two-electron singlet and a polarized triplet; coupling between the two is mediated by the difference dI ¼ ðIL IR Þ=2 of the collective nuclear spin operators of the two dots L and R, while the effective Overhauser field is given by the sum ðILz þ IRz Þ=2. In the homogeneous limit these operators obey the same algebra as the ‘‘bare’’ ones of our model, thus making our results directly applicable. 2. Achievable polarization Due to the collective coupling, singlet-like ‘‘dark states’’ can trap spin excitation and prevent cooling to the ground state. This effect is for homogeneous couplings most conveniently described in the Dicke basis [10], which is wellknown for example from the literature on superradiance. The basis states are jI; m; bi where I(I þ 1) is the eigenvalue of the collective angular momentum operator I2, and the eigenvalue of Iz is given by m. The possible values of I are N/2, N/2 1, ., the smallest value being 0 if N is even and 1/2 if N is odd; jmj I. The b is the permutation quantum number, which labels the different degenerate states with the same quantum numbers I and m. The degeneracy D of the states depends only on I and not on m, N N DðIÞ ¼ ð N=21 Þð N=2I1 Þ. When acting on a state jI; m; bi, the operator I decreases the quantum number m by one I jI; m; bi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi IðI þ 1Þ mðm 1ÞjI; m 1; bi. For m ¼ I the action of I yields zero, and thus the states jI; I; bi can not be polarized further in a system evolving according to Eq. (2). The steady state of the cooling process described by the master Eq. (2) starting from a completely mixed state is thus X DðIÞð2I þ 1ÞjI; IihI; Ij; ð3Þ rss ¼ 1=2N I
where the trace over the permutation quantum number b has been performed. We now evaluate the polarization of the steady state Eq. (3) normalized to the polarization of the ground state " # 2 4ðN 1Þ! 1 N hI z iss N ¼ 2 ; ðN=2Þ hI z i0 2N N ð½ðN 1Þ=2!Þ2 2
where for the fully polarized state one has a polarization hI z i ¼ N=2. As an example this expression can applied to the case N ¼ 2, where one finds a final polarization of 3/4. This is the expected result because one out of the four two-spin states cannot be cooled (the singlet). In Fig. 1 it is shown that for increasing particle number the possible polarization quickly decreases from 75% of the two-particle case. The negative indications for cooling purposes are obvious. pffiffiffiffiffiffi Application of the Stirling formula N!z 2pN Nþ1=2 eN gives the approximate result for N[1 hI z iss ¼ hI z i0
rffiffiffiffiffiffiffi 8 þ Oð1=NÞ: pN
ð4Þ
For the mesoscopic particle numbers of interest in the study of quantum dots, the obtainable polarization is thus negligible. In Fig. 1 it is shown that the approximate first order formula matches the exact value very well, as expected. 3. Time evolution The study of the time evolution of the nuclear spin system under the master Eq. (2) is similar to the study of the evolution of the atomic population in superradiant light emission from an ensemble of atoms. The latter problem has received considerable attention in the quantum optics literature, see e.g. the textbook [11] and review [12]. The focus of these studies was slightly different than it is in the cooling protocol we are considering. In quantum optics an ensemble with all atoms excited is the most studied situation, because for this initial condition one has pronounced signatures in the emitted light pulses and it is experimentally accessible. Here on the other hand, we are dealing with a completely mixed initial state. This complicates the situation because in this mixture superradiant as well as subradiant states occur. Nevertheless we 0.055
x 10-3
0
0.05 -0.2
Difference
a discussion of these effects). If the relevant system dynamics happens on a timescale larger than Dt, as is the case in the present study, we arrive at the continuous version of the master equation of Ref. [1] for the density matrix r of the nuclei
0.045
Coolable Fraction
966
0.04
-0.4 -0.6 -0.8 -1
0.035
-1.2
0.03
0
2000
4000
6000
8000 10000
N
0.025 Exact
0.02
Stirling 0.015
0
2000
4000
6000
8000
10000
N Fig. 1. The fraction of the nuclei that can be polarized in the case of homogeneous coupling constants. Shown are both the exact and the approximate numbers; the inset shows the difference between the two.
H. Christ et al. / Solid State Sciences 11 (2009) 965e969
where we have introduced k ¼ A2 Dt=8N 2 . The equation for the polarization hI z i does not close, but couples to the variable hI þ I i. The time evolution of this quantity depends on hI þ I z I i. Continuing this procedure leads to a hierarchy of coupled equations. Following the quantum optics literature on superradiance we make a factorization assumption z 2
2 ð5Þ ðI Þ ¼ hIz i ; hIiz Ijz i
hIiz ihIjz i.
which is equivalent to the assumption that ¼ In the following we apply this factorization to obtain analytical results for the time evolution of the polarization and check its validity by comparison with the exact numerical solution. Using this assumption the equation of motion for the polarization becomes
d z 2 hI i ¼ 2k I2 hI z i þhI z i ; dt
ð6Þ
where I2 ¼ ðI z Þ2 þ ð1=2ÞðI þ I þ I I þ Þ was used. For each given initial state jI; m; bi this equation is solved by hI z ðtÞiI;m ¼ 1=2 nI tan hðnI ð2kt c0 ÞÞ;
ð7Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where nI ¼ 4IðI þ 1Þ þ 1=2 and the initial condition depends on the value of m and enters through the constant c0 ¼ arctanh½2ðm 1=2Þ=nI =ð2nI Þ. In Fig. 2 we compare the evolution of the polarization averaged over all initial states X DðIÞhI z ðtÞiI;m ð8Þ hI z ðtÞiaveraged ¼ I;m
with the exact numerical calculation. The curves show good agreement, which serves as an indication that the factorization assumption captures the correct physics of the polarization process. To gain further insight into the evolution of the population, we consider the evolution of a pure initial state with total angular momentum and z-projection equal to the mean value of thePtotally mixed state, i.e. m ¼ 0 and Imean ¼ N=2 ð1=2N Þ I¼0 Ið2I þ 1ÞDðIÞ. Using the same approximations as for the evaluation of the total obtainable polarizations pffiffiffiffiffiffiffiffiffiffiffiffi (Eq. (4)) one gets Imean ¼ 2N=p þ Oð1Þ. In this limit the polarization develops according to 1 hI z ðtÞimean ¼ 2
! rffiffiffiffiffiffi rffiffiffiffiffiffi 8N 8N A2 Dt t ; tan h p p 8N 2
ð9Þ
0.2
0.15
extract and present here approximate explicit analytic expressions for the evolution of the nuclear spin polarization. Instead of solving the dynamics of the density matrix of the nuclei, we concentrate on the evolution of the variable of our interest, hI z i. It obeys the equation of motion
d z zd hI i ¼ tr I r ¼ 2k I þ I ; dt dt
967
0.1
Exact 0.05
Mean Approx. Mean Averaged
0
0
1
2
t/A
3
4
x 104
Fig. 2. Comparison between the various (semi-)analytical solutions based on Eq. (5) and the exact numerical result for N ¼ 100 particles. The ‘‘average’’ (dashed) line is averaged over all states in the initially totally mixed nuclear state, see Eq. (8). The two other curves are directly obtained from Eq. (7) by substituting one specific value for the angular momentum I that enters the solution as a parameter; for the dashededotted line the average value of the total angular momentum I has beenpcalculated exactly and for the dotted ffiffiffiffiffiffiffiffiffiffiffiffi curve the approximate expression I ¼ 2N=p was used.
pffiffiffiffi where we used N [1. Fig. 2 shows the evolution in the ‘‘mean state’’ for both the exact and the approximate expressions for the mean value of the average spin Imean. Using the exact mean the results are very close to the actual evolution: Both the timescale and the final polarization are correctly predicted. When the approximate expression for the mean is employed, the timescale of the process is still correct. However, the predicted final polarization differs from the real polarization. This is to be expected, since the approximated mean value of I is only correct to order 1, which translates into a Oð1=NÞ error for the polarization. From Eq. (9) we can thus extract the timescale for reaching the steady state of the polarization pffiffiffiffiffiffi N 8p ; thom ¼ A 0:1
ð10Þ
pffiffiffiffi where we used Dt ¼ 0:1 N =A, i.e. Dt is chosen such that the condition necessary for the expansion of the time evolution operator is fulfilled. 4. Microscopic description of dark states In this section a new microscopic description of dark states is introduced, which, besides providing handy intuition, will serve in Section 5 as the basis for our estimations regarding ground state cooling in homogeneous situations. We prove that one can understand all dark states I jI; I; bi ¼ 0 as a superposition of states describing n ¼ N/2 I singlets and 2I ¼ N 2n polarized spins jYi. Let us denote a singlet of spins i and j as jSij i and by jS si 2S i¼1 ; Yi the state with S singlets jSsi siþ1 i and the remaining states spins in state jYi. The
H. Christ et al. / Solid State Sciences 11 (2009) 965e969
968
idea is to show that the statistical mixture of all permutations of jS i 2S i¼1 ; Yi PI ¼
1 X y XðpÞPjSf1;.;2Sg;Yi XðpÞ N! p˛SN
(where for a permutation p ˛ SN let XðpÞjSfsi g; Yi ¼ jSfpðsi Þg; Yi) is equal to the projector on the the (I, I )subspace:
ith Hadamard mode could be achieved, e.g. by performing phase flips jYij / jYij for all j with (Hm)ji ¼ 1. When applied after cooling with the first mode, such a mode change has no effect on nuclei already in state jYi but singlets are transformed to triplets (with zero z-projection) if the phase of only one of the two spins is flipped, so that the state in the new mode is X
pðnÞ
n
To see this, we first observe that SN Hp 1XðI;MÞ ðpÞ˛M2N with X(I, M )(p) acting (as above) on the space spanned by the vectors jI; M; bi with b ¼ 1; .; DðIÞ a label for the eigen ! values of a set of permutation operators extending I 2 ; and Iz to a complete commuting set (e.g. P2i1;2i ; i ¼ 1; .; N=2 cf. [10]) is a representation of the permutation group SN. The representation must be irreducible, since if it were not, there would exist at least two sets of vectors in HðI;MÞ that are not connected by SN, hence the projector on one of those would ! be an operator commuting with f I 2 ; I z ; P2i1;2i g, contradicting that this is a complete set of commuting observables on ðC2 Þ5N . Finally, we clearly have XI,I(p)PIX(I,I )(p)y ¼ PI and by Schur’s Lemma (which states that any operator commuting with all elements of a irrep is proportional to the identity see, e.g. Ref. [13]) it follows that PI f1I;I . Counting the number of permutations respectively the degeneracy of the (I, I )-subspace gives the normalization factor. This shows that the projector on the (I, m)-subspace is given by the equal-weight mixture of all permutations of N/2 I singlets and the remaining 2I spins in the state jI; mi. 5. Mode changes To cool beyond the dark state, the coupling operator I needs to be changed. In view of the analogy between a (collectively addressed) system of spins and a bosonic mode, addressing a differsuch a transformation of I can be viewed asP ent ‘‘mode’’ of the system [1]. Any ~I ¼ i gi Ii with ! g not parallel to ð1; .; 1Þ represents a different mode in this terminology. The number of modes that are needed to cool the nuclear spins to the ground state is in general not easily estimated due to the fact that orthogonal spin modes do not commute [14]. In order to get an estimate for the number of modes needed for cooling the spins, we use the simple Hadamard basis for the modes. The Hadamard matrices are recursively defined by H0 ¼ 1, and 1 Hm Hm : Hmþ1 ¼ pffiffiffi 2 Hm Hm The size of the matrix is 2m. Its columns are mutually orthogonal and represent the different modes we aim to address. Changing from the standard homogeneous coupling to the
qðn;uÞXðpÞPjS1;.;2ðnuÞ;Yi XðpÞy 5jT0 ihT0 j5u
p˛SN u¼0
where q(n, u) denotes the probability that u of the n singlets become a triplet after the mode change. We assume that changing from one Hadamard mode to the next typically breaks up half the singlets in each of the terms in the mixture, i.e. u ¼ n=2. We now calculate the total angular momentum I2 of this state, by first noting that in each of the above terms in the mixture one can split the collective operators up into terms that act on spins which remained untransformed by the mode change, and the rest Sall spins ¼ ðSin some T0 þ Srest Þ. The mixed terms between the two sums can be seen to be 0 readily. The terms involving the unchanged (rest) spins have the same total spin as before the mode change: Only singlets have been taken away. For the spins that have turned from singlets to triplets one can easily verify that their contribution to the new total angular momentum is (N/2 Iold)/4. In a semiclassical treatment we now calculate in the way outlined above the average angular momentum after k mode changes and arrive at the recursive formula Ikþ1 ðIkþ1 þ 1Þ ¼ Ik ðIk þ 1Þ þ ðN=2 Ik Þ=4;
ð11Þ
where k is the index for the number of modes changes, and from which we extract the minimal compatible polarization, Izk ¼ Ik. The numerical solution of the above recursive formula presented in Fig. 3 indicates that order of OðNÞ modes
0.8 0.7 0.6
1 X
I; I; b I; I; b : PI ¼ DðIÞ b
n XX
0.5 0.4 0.3 0.2 0.1 0
0
500
1000
1500
2000
# modes k Fig. 3. Estimate of the achieved cooling after a given number of (Hadamard-) mode changes for N ¼ 103 nuclear spins.
H. Christ et al. / Solid State Sciences 11 (2009) 965e969
are necessary to completely cool the nuclear spins. We have verified this scaling behavior for a wide range of particle numbers. Note that a scheme employing Hadamard modes is not optimal: if one had complete freedom in the choice of modes, one could cool each spin individually and therefore cool everything to the perfect ground state in N steps; which is already better than the cooling procedure shown in Fig. 3. We are thus lead to the conclusion that achieving high polarization for strictly homogeneous couplings is hard: Sophisticated and undesirable control mechanisms for the electron wave function might be needed. As seen in Ref. [1], an inhomogeneous coupling will significantly simplify the cooling process.
6. Conclusions We have presented a study of nuclear spin cooling in quantum dots in the limit of a homogeneous electron wave function. We calculated exactly the cooling potential, presented analytical formulas for the time evolution of the polarization and, after introducing a handy intuition for the microscopic description of the dark states in the nuclear ensemble, showed that many different (Hadamard-) electron wave function profiles are needed to completely polarize the nuclei. The results underline the importance of the inhomogeneous Knight shift in nuclear spin cooling, which mitigates these limitations so strongly, that polarizations above 90% are achievable in realistic setups [1].
969
References [1] H. Christ, J.I. Cirac, G. Giedke, Phys. Rev. B 75 (2007) 155324. [2] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120 cond-mat/ 9701055. [3] J.M. Taylor, G. Giedke, H. Christ, B. Paredes, J.I. Cirac, P. Zoller, M.D. Lukin, A. Imamo˘glu, 2004, cond-mat/0407640. [4] D.H. Feng, I.A. Akimov, F. Henneberger, Phys. Rev. Lett. 99 (2007) 036604 cond-mat/0703386. [5] P. Maletinsky, A. Badolato, A. Imamoglu, Phys. Rev. Lett. 99 (2007) 056804. [6] A. Imamo˘glu, E. Knill, L. Tian, P. Zoller, Phys. Rev. Lett. 91 (2003) 017402 cond-mat/0303575. [7] S.I. Erlingsson, Y.V. Nazarov, Phys. Rev. B 70 (2004) 205327. [8] J. Baugh, Y. Kitamura, K. Ono, S. Tarucha, Phys. Rev. Lett. 99 (2007) 096804. [9] J.R. Petta, J.M. Taylor, A.C. Johnson, A. Yacoby, M.D. Lukin, C.M. Marcus, M.P. Hanson, A.C. Gossard, 2007, arXiv:0709.0920. [10] F.T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, Phys. Rev. A 6 (1972) 2211. [11] A.V. Andreev, V.I. Emelyanov, Y.A. Ilinski, Cooperative Effects in Optics: Superradiance and Phase Transitions, IOP Publishing, London, 1993. [12] M. Gross, S. Haroche, Phys. Rep. 93 (1982) 301. [13] T. Inui, Y. Tanabe, Y. Onodera, Group Theory and its Applications in Physics, second ed, Springer, Berlin, 1996. [14] In the bosonic approximation [1] the number of required modes is easily seen to be N, because the modes are mutually independent and initially all N modes are occupied and consequently have to be cooled one by one (neglecting the effects of inhomogeneous Knight fields). The bosonic model is only exact for high polarization, and in general is expected to be a lower bound on the performance of the cooling protocol for the spins. It is unclear, if the scaling behavior of the number of required modes is the same pffiffiffiffifor spins. One might think, for example, that one needs pffiffiffiffi only Oð N Þ modes in the spin case, because in the first mode w N excitations can be removed from the spin system.