Protecting and Enhancing Spin Squeezing via Continuous Dynamical Decoupling Adam Zaman Chaudhry1 and Jiangbin Gong1, 2, ∗
arXiv:1203.4631v1 [quant-ph] 21 Mar 2012
1
NUS Graduate School for Integrative Sciences and Engineering, Singapore 117597, Singapore 2 Department of Physics and Center for Computational Science and Engineering, National University of Singapore, Singapore 117542, Singapore
Realizing useful quantum operations with high fidelity is a two-task quantum control problem wherein decoherence is to be suppressed and desired unitary evolution is to be executed. The dynamical decoupling (DD) approach to decoherence suppression has been fruitful but synthesizing DD fields with certain quantum control fields may be experimentally demanding. In the context of spin squeezing, here we explore an unforeseen possibility that continuous DD fields may serve dual purposes at once. In particular, it is shown that a rather simple configuration of DD fields can suppress collective decoherence and yield a 1/N scaling of the squeezing performance (N is the number of spins), thus making spin squeezing more robust to noise and much closer to the so-called Heisenberg limit. The theoretical predictions should be within the reach of current spin squeezing experiments. PACS numbers: 42.50.Dv, 03.67.Pp, 03.65.Yz, 03.75.Gg
The feasibility of “dynamical decoupling” (DD) [1] in effectively isolating a quantum system from its environment has attracted great theoretical and experimental interests. Remarkable progress towards efficient protection of quantum states has been achieved [2–5]. In contrast, high-fidelity protection of quantum operations (such as quantum gates or quantum metrology schemes) is more challenging experimentally. As decoherence must be suppressed during quantum operations, it is natural to synthesize DD fields with other fields implementing a desired quantum operation. However, this bottom-up approach may require complicated coherent control fields. For example, explicit solutions to dynamically corrected quantum gates are sophisticated [6–8], and even a simple quantum metrology protocol, when combined with DD, already becomes a rather involving practice [9]. A top-down approach to the protection of useful quantum operations should be a worthy direction, along which we aim to better exploit the system’s own Hamiltonian under DD fields. In essence we are faced with a two-task problem: decoherence is to be suppressed and desired (almost) unitary evolution is to be executed. Is it possible to directly construct DD fields serving the dual tasks at once? Motivated by recent studies of decoherence effects on spin squeezing [10–17] and by recent exciting experiments of spin squeezing [18, 19], here we use the spin squeezing context to give a positive answer to our question. That is, it is feasible to protect and enhance spin squeezing at the same time by searching for a special configuration of DD fields. The dual roles of DD fields arise from two facts: (i) DD fields modulate both systembath interaction and the system Hamiltonian itself that describes the spin-spin interaction, and (ii) the system Hamiltonian itself under the modulation of DD fields may generate more useful quantum evolution. We emphasize that the enhancement in spin squeezing we achieve is not a secondary outcome of decoherence suppression.
Rather, the enhanced spin squeezing is far superior to what can be normally achieved under “decoherence-free” conditions. Indeed, we predict a 1/N scaling of the obtained spin squeezing performance, where N is the number of spins. Our theoretical results should be testable by modifying existing experiments. In addition, because spin squeezing is closely related to multi-partite entanglement [20, 21], the results are also of interest to ongoing studies of entanglement protection [22–26]. Consider then a collection of N identical spins (or qubits). In terms of the standard Pauli matrices, the dynamics can be described by collective angular momenP (m) tum operators Jk = 12 N m=1 σk , with k = x, y, z. In spin squeezed states, quantum fluctuations of the collective angular momentum in one direction are significantly reduced at the price of increased uncertainty in another direction [27, 28], thus offering higher precision in quantum metrology [29, 30]. For instance, we can improve high-precision spectroscopy and atomic clocks which are currently limited by spin noise [31]. Using the angular momentum commutation relations, the two most widely used measures of spin squeezing, ξS2 2 [27] and ξR [29], are found to be bounded by 1/N [32]. This fundamental limit to the amount of spin squeezing achievable reflects the Heisenberg precision limit in quantum measurement [28, 33]. In practice, the achievable degree of squeezing is considerably worse than the 1/N limit for two main reasons. First, squeezing is in general degraded by decoherence or noise [34]. The environment tends to destroy squeezing, causing the sudden death of squeezing [13]. It may also change the optimal squeezing time window in an unpredictable way, leading to a non-optimal squeezing generation [19]. Second, Hamiltonians that can be implemented so far cannot reach the 1/N scaling in theory. For instance, in two recent experiments [18, 19] based on two-mode Bose-Einstein condensate (BEC), the so-called one-axis twisting (OAT) Hamil-
2 tonian [27] HOAT = χJx2 is realized, which can at most 2 2 generate ξR ∼ 1/N 3 , not to mention decoherence effects. It is thus clear that protecting spin squeezing against decoherence and pushing spin squeezing towards the 1/N scaling would be of wide interest. We start by considering a OAT system interacting with an environment, with the total Hamiltonian modeled by H = H0 + HB + HSB , where, H0 = HOAT = χJx2 , HB is the Hamiltonian of the environment, and HSB represents the system-environment coupling. HSB is assumed to be HSB = Bx Jx + By Jy + Bz Jz ,
(1)
where the Bk are arbitrary bath operators (or randomly fluctuating noise for a classical bath). Though coupling terms that are nonlinear in Jk are not considered here, the HSB in Eq. (1) is already quite general insofar as it covers a broad class of problems with both dephasing and relaxation. There is a standard route to seek a control Hamiltonian Hc (t) that can effectively average out HSB and hence suppress decoherence. In particular we consider a continuous Hc (t) of period tc , whose time-ordered exponential Rt defines a unitary operator Uc (t) = T exp[−i 0 Hc (t′ )dt′ ] (~ = 1 throughout), with Uc (t+tc ) = Uc (t). The Magnus expansion [35, 36] indicates that if Z tc Uc† (t)HSB Uc (t) dt = 0, (2) 0
then to its first order HSB is suppressed. Extending previous studies for single-qubit and two-qubit systems [8, 37, 38], we choose Uc (t) = e−2πiny Jy t/tc e−2πinx Jx t/tc ,
(3)
where nx and ny are non-zero integers. For any nx 6= ny , Uc (t) in Eq. (3) satisfies the first-order DD condition of Eq. (2). Qualitatively, such Uc (t) causes the collective angular momentum operators to rapidly rotate in two independent directions and as a result, HSB is averaged out to zero. Using i dUc (t)/dt = Hc (t)Uc (t), we obtain the following DD control Hamiltonian, Hc (t) = ωny Jy + ωnx [Jx cos(ωny t) − Jz sin(ωny t)], (4) where ω ≡ 2π/tc . The total system Hamiltonian Hs (t) = HOAT + Hc (t) then becomes
example a time-dependent Zeeman shift. The Jx and Jy terms can be generated by use of electric-dipole interaction - considering a circularly polarized transition, a constant electric field along y direction and an oscillating field along x direction lead to the desired Jx and Jy terms [39]. With a continuous control Hamiltonian Hc (t) implemented, decoherence can be well suppressed for sufficiently large ω. Two observations are in order. First, as shown in Eq. (4), infinite DD solutions with different (nx ,ny ) combinations are found. Second, the control Hamiltonian averages out HSB via fast modulations of Jk , so the system’s self-interaction term Jx2 is necessarily modulated at the same time. One opportunity is then emerging: among all the DD solutions, can we identify a particular type that modulate the system’s self-interaction Hamiltonian in a useful manner so as to enhance squeezing while suppressing decoherence? With the system decoupled from its environment, it can be shown that the system evolution operator is given ¯ by Us (t) ≈ Uc (t)e−iHt [36], where the time-averaged ¯ is found to be Hamiltonian H Z tc ¯ = χ Uc† (t)Jx2 Uc (t) dt. (5) H tc 0 A straightforward though rather tedious calculation yields Uc† (t)Jx2 Uc (t) 1 = sin(2ωnx t) sin2 (ωny t)[Jz Jy + Jy Jz ] 2 1 + sin(2ωny t) cos(ωnx t)[Jx Jz + Jz Jx ] 2 1 + sin(2ωny t) sin(ωnx t)[Jx Jy + Jy Jx ] 2 + Jx2 cos2 (ωny t) + Jy2 sin2 (ωnx t) sin2 (ωny t) + Jz2 cos2 (ωnx t) sin2 (ωny t).
(6)
Using Eq. (6), one finds the time-averaged Hamiltonian ¯ has two different forms. Specifically, if nx 6= 2ny , H ¯ = χ Jx2 (up to a constant), which is just the original H 4 OAT Hamiltonian with the nonlinear coefficient scaled down by a factor of four. If nx = 2ny , which we call the “double-resonance” (DR) condition, we obtain (up to a constant)
Hs (t) = χJx2 + ωny Jy + ωnx [Jx sin(ωny t) − Jz cos(ωny t)].
¯ =H ¯ DR = χ Jx2 + Jx Jy + Jy Jx . H 4
To elaborate how Hs (t) can be realized, we rotate the coordinate system along the y-axis by π/2, transforming Hs (t) to Hs′ (t) = χJz2 + ωny Jy − ωnx [Jx sin(ωny t) + Jz cos(ωny t)]. We now comment on each term of Hs′ (t). The first Jz2 term describes spin-spin interaction, as is realized in experiments [18, 19]. The last term linear in Jz can be realized by an oscillating energy bias using for
¯ DR is seen to be a mixture of a OAT Remarkably, H Hamiltonian and a well-known two-axis twisting (TAT) Hamiltonian HTAT = χ(Jx Jy + Jy Jx ) [27, 28]. Since HTAT is known to produce the best scaling of squeezing, we are motivated to examine the squeezing performance ¯ DR , naturally obtained by one type of DD fields to of H fight against both relaxation and dephasing.
(7)
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We consider an initial state |J, −Ji describing all spins “pointing down”, that is, an eigenstate of Jz with eigen¯ DR and invalue −N/2. To verify our expressions of H vestigate its potential benefits we first switch off HSB and evolve the wavefunction numerically under Hs (t) = 4 min(∆J
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n ~⊥ HOAT + Hc (t). We use ξS2 ≡ [27] to quantify N squeezing, where ~n⊥ denotes a direction perpendicular to the mean spin direction and the minimum is taken over all such directions. The dynamical behavior of ξS2 is found 2 2 ~ 2 ), so to be essentially the same as ξR (ξR = ξS2 (J/|hJi|) 2 only the behavior of ξS is presented below. We set χ = 1 and tc = tmin /Ncyc , where tmin is the optimal squeezing
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FIG. 1. (color online) Spin squeezing measure ξS2 against time ¯ DR (solid, t using the OAT Hamiltonian (dot-dashed, red), H dark blue), and Hs (t) with Ncyc = 5 (dotted, magenta) and Ncyc = 20 (dashed, blue) for N = 10 (i.e., J = 5). Note that the dynamics generated by Hs (t) with Ncyc = 20 are al¯ DR . most indistinguishable from the dynamics generated by H Here we use nx = 2 and ny = 1, and tmin was found to be approximately 0.491.
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FIG. 2. (color online) Same as in Fig. 1, but this time we have N = 100 (i.e., J = 50) with Ncyc = 10 (dotted, magenta), Ncyc = 30 (dashed, blue), and tmin ≈ 0.0909. It is also observed here that the dynamics generated by Hs (t) with Ncyc = 30 are well captured by the dynamics generated ¯ DR (solid, dark blue). by H
FIG. 3. (color online) For J = 5, spin squeezing measure ξS2 against time t using the bare OAT Hamiltonian without noise ¯ DR (solid, dark blue), the OAT Hamil(dot-dashed, red), H tonian with noise but without DD fields (dashed, magenta), and the OAT Hamiltonian in the presence noise and the DD fields with nx = 2, ny = 1, tmin ≈ 0.491, and Ncyc = 20 (dotted blue line, which is almost on top of the solid line). An average over 2000 sample paths of the noise was taken. The noise parameters are α = 2 and σ 2 = 20.
¯ DR , and Ncyc is a positive integer. In Fig. 1, the time for H squeezing performance of Hs (t) is compared with that of HOAT = χJx2 , for N = 10. It is seen that Hs (t) generates much better squeezing - the minimum value of ξS2 using Hs (t) is approximately 0.15, but we can only achieve a value of approximately 0.2 using HOAT . In addition, it is seen from Fig. 1 that the dynamics under Hs (t) is indeed well captured by the dynamics under the time-averaged ¯ DR . In particular, within each period of Hamiltonian H ¯ DR yield some fluctuating differences in tc , Hs (t) and H ξS2 , but at integer multiples of tc when Uc (t) = 1, excellent agreement between them is observed. Further, if we reduce tc by increasing the strength and the frequency of the periodic control Hamiltonian Hc (t), the fluctuating differences become smaller. This reflects the fact that ¯ DR is obtained under a first-order approximation. We H next increase N tenfold and essentially the same results are obtained in Fig. 2, but with the advantage gained by our DD fields displayed even more evidently. For example, in Fig. 2 it is seen that the minimum values of ξS2 using Hs (t) and HOAT are approximately 0.019 and 0.048, respectively - so more than a two-fold improvement can therefore be obtained. Other calculations indicate that ¯ DR goes down with fixed for larger N , the accuracy H tc . Thus, a higher driving frequency ω would be more favored as N increases. Having confirmed that the double-resonance condition nx = 2ny is useful for spin squeezing, let us now turn to the full problem by switching on the system-environment coupling. For convenience we model Bx , By and Bz in Eq. (1) as three independent Gaussian colored noise processes, with the same inverse correlation time α and noise
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FIG. 4. (color online) Same as in Fig. 3, but now with J = 50, tmin ≈ 0.0909, Ncyc = 30, and σ 2 = 100. An average over 100 sample paths of the noise was taken. Similar to what is observed in Fig. 3, the squeezing performance of the OAT Hamiltonian in the presence of noise and DD fields (dotted, blue) is much better than that of a bare OAT Hamiltonian in the absence of noise (dot-dashed, red).
of Fig. 5 for a comparison of two different scalings]. The DD fields under the double-resonance condition hence allows squeezing to occur in the presence of noise and in the mean time brings about a squeezing enhancement factor of N 1/3 , which is in principle unlimited as N increases. It is also interesting to note how this work differs from a recent proposal for realizing TAT Hamiltonian by applying a designed pulse sequence to a OAT Hamiltonian [40]. While our starting point is continuous DD fields for decoherence suppression, the short control pulses considered in Ref. [40] do not average out HSB in Eq. (1) to zero. ¯ DR found here unFurther, the effective Hamiltonian H der a double-resonance condition is a mixture of OAT ¯ DR is a and TAT Hamiltonians. To our knowledge, H newly found, physically motivated Hamiltonian that can generate the ξS2 ∼ 1/N scaling. -18 -20 ξS2 (in dB)
variance σ 2 . We numerically compute the dynamics of squeezing for Hs (t) in the presence of noise and then compare it with that generated by HOAT , with noise or without noise. As shown in Figs. 3 and 4, Hs (t) with noise yields much better squeezing than HOAT with noise. This may be understood as an outcome of decoherence suppression. On the other hand, Hs (t) with noise also generates better squeezing than HOAT in the absence of noise. Hence our DD fields have played one more role in addition to decoherence suppression. Note also that the time to obtain maximum squeezing is in excellent agree¯ DR , hence avoiding dement with that obtained from H coherence effects on the optimal squeezing time and also ¯ DR in predicting the confirming again the usefulness of H optimal squeezing time. Having shown how DD fields may suppress decoherence and enhance the spin squeezing generation as a unitary process, we finally investigate how the squeezing ¯ DR scales with N . Within the validperformance of H ¯ DR as an effective Hamiltonian (for deity regime of H scribing the dynamics associated with the OAT Hamiltonian in the presence of noise and continuous DD fields), ¯ DR with the scaling of the squeezing performance of H N represents to what degree our DD fields can protect and enhance spin squeezing. Calculations for even larger values of N then become necessary. We first compare ¯ DR with what is known to give the performance of H the best scaling behavior, namely, the TAT Hamiltonian ¯ DR HTAT = χ(Jx Jy + Jy Jx ). Significantly, although H produces slightly less squeezing than HTAT , two close and parallel lines describing their respective performance are seen in Fig. 5, indicating that both cases give the ξS2 ∼ 1/N scaling. By contrast, Fig. 5 also presents the 1/N 2/3 scaling of the OAT Hamiltonian [also see the inset
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FIG. 5. (color online) Minimum value of ξS2 - in dB, defined as 10 log(ξS2 ) - plotted against N (log scale) for dynamics generated by the OAT Hamiltonian (upper, dot-dashed, red), the TAT Hamiltonian (lower, dashed, magenta), and the time¯ DR (middle, solid, green). In the inaveraged Hamiltonian H set, the upper (dot-dashed, red) line has been transported ¯ DR vertically to show clearly the better scaling behavior of H 2 and the TAT Hamiltonian. The behavior of ξR (not shown) is similar.
To conclude, by considering a class of continuous fields to suppress both dephasing and relaxation in the dynamics of spin squeezing, we are able to identify a special type of DD solutions that can effectively yield a previously unknown spin squeezing Hamiltonian, generating the 1/N scaling of squeezing performance in the presence of an environment. With their dual roles in decoherence suppression and in generating more useful quantum evolution identified, the found DD fields are appealing from an experimental point of view. Our results should be able to help design new experimental studies of spin squeezing based on one-axis twisting Hamiltonians (such as those using two-mode BEC). Indeed, by exploiting system’s own spin-spin interaction Hamiltonian under the modulation of continuous DD fields, we expect to see other
5 interesting DD designs that can carry out desired quantum operations while protecting quantum coherence. A.Z.C. would like to thank Derek Ho and Juzar Thingna for helpful discussions.
∗
[1] [2] [3] [4] [5] [6] [7]
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
[email protected] L. Viola and S. Lloyd, Phys. Rev. A 58, 2733 (1998). G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007). W. Yang et al., Frontiers of Physics 6, 2 (2011). M. J. Biercuk et al., Nature (London) 458, 996 (2009). J. Du et al., Nature (London) 461, 1265 (2009). L. Viola et al., Phys. Rev. Lett. 83, 4888 (1999). K. Khodjasteh and L. Viola, Phys. Rev. Lett. 102, 080501 (2009); K. Khodjasteh et al., Phys. Rev. Lett. 104, 090501 (2010). A. Z. Chaudhry and J. B. Gong, Phys. Rev. A 85, 012315 (2012). X. Rong et al., EPL 95, 60005 (2011). C. Simon and J. Kempe, Phys. Rev. A 65, 052327 (2002). A. Andre and M. D. Lukin, Phys. Rev. A 65, 053819 (2002). J. K. Stockton et al., Phys. Rev. A 67, 022112 (2003). X. Wang et al., Phys. Rev. A 81, 022106 (2010). Y. Li et al., Phys. Rev. Lett. 100, 210401 (2008). A. Sinatra et al., Phys. Rev. Lett. 107, 060404 (2011). Nir Bar-Gill et al., Phys. Rev. Lett. 107, 010404 (2011). G. Watanabe and H. Makela, Phys. Rev. A 85, 023604 (2012). C. Gross et al., Nature (London) 464, 1165 (2010). M. F. Riedel et al., Nature (London) 464, 1170 (2010). O. Guhne and G. Toth, Physics Reports 474, 1 (2009).
[21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
[31] [32]
[33] [34]
[35] [36] [37] [38] [39]
[40]
L. Amico et al., Rev. Mod. Phys. 80, 517 (2008). M. Mukhtar et al., Phys. Rev. A 81, 012331 (2010). M. Mukhtar, et al., Phys. Rev. A 82, 052338 (2010). Y. Wang et al., Phys. Rev. Lett. 106, 040501 (2011). L. Jiang and A. Imambekov, Phys. Rev. A 84, 060302 (2011). Z. Y. Wang and R. B. Liu, Phys. Rev. A 83, 022306 (2011). M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993). J. Ma et al., Physics Reports 509, 89 (2011). D. J. Wineland et al., Phys. Rev. A 50, 67 (1994). In the prototype Ramsey interferometry setup, the phase √ uncertainty using uncorrelated particles is ∆φ = 1/ N , while using a squeezed √ state of N particles, the uncertainty is ∆φ = ξR / N , where ξR is a measure of spin squeezing. G. Santarelli et al., Phys. Rev. Lett. 82, 4619 (1999). 2 The squeezing measures, ξS2 and ξR , basically measure the spin fluctuations, so a lower value implies a more squeezed state. V. Giovannetti et al., Science 306, 1330 (2004). We should also note that the environment can also induce squeezing (see Refs. [16, 17]). However, in such cases, one assumes that the environment affects the system in a particular way. For instance, in Ref. [17], a particular jump operator in the Lindblad equation modeling the system dynamics is assumed. W. Magnus, Commun. Pure Appl. Math 7, 649 (1954). P. Facchi et al., Phys. Rev. A 71, 022302 (2005). F. F. Fanchini et al., Phys. Rev. A 75, 022329 (2007). F. F. Fanchini and R. d. J. Napolitano, Phys. Rev. A 76, 062306 (2007). S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University Press, Oxford, U.K., 2006). Y. C. Liu et al., Phys. Rev. Lett. 107, 013601 (2011).
