Seeing Scrambled Spins

Seeing Scrambled Spins VIEWPOINT Two experimental groups have taken a step towards observing the ``scrambling'' of information that occurs as a many-body quantum system thermalizes. by Brian Swingle and Norman Y. Yao hysicists have long wondered whether and how iso- lated quantum systems thermalize—questions that are relevant to systems as diverse as ultracold atomic P gases and black holes. Recent theoretical and ex- perimental advances are bringing fresh insight into this line of inquiry. At one extreme, researchers have shown that disorder can fully arrest thermalization in certain isolated many-body quantum systems [1]. At the other extreme, Figure 1: A classical chaotic system can be diagnosed by the surprising results from the field of quantum gravity have presence of the buttery effect, in which a small perturbation like established that black holes are, in some sense, the fastest the tiny ap of a buttery's wing has a huge effect on the system at thermalizers in nature [2–4]. A common thread running some later point in time. (Left) Another version of the classical through these developments is an emerging focus on the dy- buttery effect compares the situations of running time forward (blue line) with running it backward after the buttery is still (white) namics of quantum information, in which thermalization is or after the buttery aps its wings (red). Without the buttery ap, associated with “scrambling,” or the loss of accessible infor- the system returns to its initial state; with it, the state of the system mation. Two groups, one in China [5] and one in the US [6], eventually differs drastically from its initial state. (Right) Li et al. [5] have taken a step towards tracking this scrambling of infor- and Gärttner et al. [6] performed an analogous experiment with mation in systems of quantum spins. quantum spin systems, here described by a wave function Y. Both groups used quantum-control techniques to evolve their systems The lore of thermalization goes as follows. Suppose you forward in time (blue line), to apply a perturbation W, and to evolve initialize a collection of quantum spins into one of two dis- the systems backward in time (red line). They then performed a tinct configurations. Now couple the system to a large heat measurement of V to diagnose the effect of the perturbation. bath. After equilibrium is reached, the final state of the spins (APS/Alan Stonebraker) will be independent of the spins’ initial configuration. In other words, information about the initial state of the spins has been irrevocably lost to the bath. radiation? Since a black hole is fundamentally a thermal ob- ject, this paradox is intimately related to how information But thermalization does not require a bath to proceed. In dynamics leads to thermalization. Specifically, one could a complex many-body quantum system, information about imagine that when something falls into a black hole, the in- the initial state may instead be “hidden” in elaborate corre- formation about it is encoded—albeit in scrambled form—in lations among the system’s constituents. The information in the radiation emitted during evaporation. such a scrambled state is not lost, because the final state can be related to the initial state by a unitary transformation. But Experiments that can probe the quantum dynamics of it may be inaccessible to any reasonable local measurement. black holes are currently out of reach. But scrambling is also relevant to isolated collections of strongly interacting atoms, The concept of information scrambling first arose in at- ions, molecules, and photons—all systems that physicists tempts to understand the black hole information paradox, can prepare in the lab. As a bonus, it may be possible to which asks: How can information about what fell into a engineer Hamiltonians in these systems that scramble infor- black hole be both trapped inside the event horizon and lib- mation as fast as black holes. The most direct way to detect erated as the black hole “evaporates” by emitting Hawking scrambling would be to measure a system’s entropy over time, though this is typically too hard to do. Instead, re- Department of Physics, University of Maryland, College Park, MD searchers have figured out that they can partially diagnose scrambling using unusual correlation functions called out- 20740, USA Department of Physics, University of California, Berkeley, CA of-time-order correlators (OTOCs) [2, 3, 7]. These correlators 94720, USA effectively involve a many-body “time machine.” Given two physics.aps.org 2017 American Physical Society 19 July 2017 Physics 10, 82 simple quantum operators W and V, one imagines compar- systems is well beyond what physicists can simulate on ing two processes: (i) Evolve the system forward in time, a classical computer. However, the team confined its ex- apply W, evolve backward in time, and apply V; (ii) apply periment to the dynamics of a more tractable subspace of V, evolve forward in time, apply W, and evolve backward quantum states. Moreover, despite the large number of spins in time. in their experiment, the spin Hamiltonian that they engi- neered was not chaotic, and their measurements of OTOCs, What does this comparison tell you? Drawing on an anal- like those of the molecular spin experiment, took place far ogy to classical chaos, one interpretation is that comparing from the limit of many-body scrambling. the two processes reveals the sensitivity of a measurement While neither experiment reaches the limit of true many- of V to a perturbation W—say, a kick from an external body chaos, both raise crucial questions. Can one dis- field—that happened some time in the past. If the mea- tinguish information that is scrambled from that which is surement is very sensitive to the perturbation, we have a simply lost because of environmental noise and spin deco- quantum version of the classical butterfly effect, in which herence? Can one correct for small errors that result from a small initial perturbation eventually has a major effect imperfectly evolving a system backward in time? Using the (Fig. 1). Taking the analogy further, a quantum system in rich data set from their ion experiment, Gärttner et al. were which information becomes scrambled can be viewed as a able to explore and model various sources of such imperfec- quantum chaotic system, and the OTOC provides a measure tion such as magnetic-field noise. of the scrambling. Quantum thermalization is a rapidly developing field. In Unfortunately, measuring OTOCs is difficult. Existing fact, two new scrambling experiments appeared just recently proposals [8–10] for doing so require the experimenter ei- [11, 12]. The near future promises experiments of increas- ther to evolve a system forward in time under a Hamiltonian ing complexity—both larger system sizes and more chaotic and then backwards in time by implementing the negative of Hamiltonians. Building on the work by Li et al. and Gärttner this Hamiltonian or to make a delicate comparison of two et al., it seems likely that experiments will soon forge beyond many-body quantum states. Thanks to the growing tool- what computers can simulate, revealing the dynamics of in- box of quantum-control techniques, these difficult tasks are formation scrambling in previously inaccessible regimes. now (somewhat) possible and the teams from China and the US have demonstrated proof-of-principle measurements of OTOCs. This research is published in Physical Review X and in Nature Jun Li, from the Beijing Computational Science Research Physics. Center, and colleagues used four nuclear spins in the iodotri- fluroethylene molecule [5]. After preparing the spins in a REFERENCES particular initial state, they applied a sequence of control pulses to engineer a quantum simulation of the mixed-field [1] R. Nandkishore and D. A. Huse, ``Many-Body Localization and Ising Hamiltonian, evolving this Hamiltonian forward in Thermalization in Quantum Statistical Mechanics,'' Annu. Rev. time. After perturbing the spins, they used another series of Cond. Matt. Phys. 6, 15 (2015). control pulses to implement the negative of the Ising Hamil- [2] S. H. Shenker and D. Stanford, ``Black Holes and the Buttery tonian, thus enabling the necessary “rewinding” of time, and Effect,'' J. High Energy Phys. 2014, 67 (2014). again evolved the spins. Their measurement of the final spin [3] A. Kitaev, Talk at Fundamental Physics Prize Symposium Nov. 10, 2014. state effectively yields the OTOC. But while their Hamilto- [4] J. Maldacena, S. H. Shenker, and D. Stanford, ``A Bound on nian is, in principle, chaotic, the system size is so small that Chaos,'' J. High Energy Phys. 2016, 106 (2016). its full evolution can be directly simulated on a computer, [5] J. Li, R. Fan, H. Wang, B. Ye, B. Zeng, H. Zhai, X. Peng, and it is far from the limit of many-body chaos. and J. Du, ``Measuring Out-of-Time-Order Correlators on a Nu- Martin Gärttner, from the University of Colorado Boulder, clear Magnetic Resonance Quantum Simulator,'' Phys. Rev. X JILA, and the National Institute of Standards and Technol- 7, 031011 (2017). ogy (all in Boulder, Colorado) [6], and colleagues studied [6] M. Gärttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. the dynamics of a much larger system consisting of more J. Bollinger, and A. M. Rey, ``Measuring Out-of-time-order 9 + Correlations and Multiple Quantum Spectra in a Trapped-ion than one hundred Be ions confined in a two-dimensional Quantum Magnet,'' Nat. Phys. (2017). electromagnetic trap. The valence electron spin of each ion [7] A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 55, behaves as an S = 1/2 magnetic moment. The Boulder 2262 (1969), [Sov. Phys. JETP 28, 1200 (1965)]. team implemented a long-range classical Ising Hamiltonian [8] B. Swingle, B. Bentsen, M. Schleier-Smith, and P. Hayden, by using a laser to couple the spins to the motional modes ``Measuring the Scrambling of Quantum Information,'' Phys. of the ion crystal. Then, using a protocol analogous to that Rev. A 94, 040302 (2016). of the team from China, they evolved the system “forward” [9] N. Y. Yao, F. Grusdt,, B. Swingle, M. D. Lukin, D. M. Stamper- and “backward” in time to measure the OTOC. The gen- Kurn, J. E. Moore, and E. A. Demler, ``Interferometric Approach eral dynamical evolution of one hundred two-level quantum to Probing Fast Scrambling,'' arXiv:1607.01801. physics.aps.org 2017 American Physical Society 19 July 2017 Physics 10, 82 [10] G. Zhu, M. Hafezi, and T. Grover, ``Measurement of Many-Body [12] E. J. Meier, J. Ang'ong'a, F. A. An, and B. Gadway, ``Exploring Chaos Using a Quantum Clock,'' Phys. Rev. A 94, 062329 Quantum Signatures of Chaos on a Floquet Synthetic Lattice,'' (2016). arXiv:1705.06714. [11] K. X. Wei, C. Ramanathan, and P. Cappellaro, ``Exploring Lo- calization in Nuclear Spin Chains,'' arXiv:1612.05249. 10.1103/Physics.10.82 physics.aps.org 2017 American Physical Society 19 July 2017 Physics 10, 82 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physics American Physical Society (APS)

Seeing Scrambled Spins

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Abstract

VIEWPOINT Two experimental groups have taken a step towards observing the ``scrambling'' of information that occurs as a many-body quantum system thermalizes. by Brian Swingle and Norman Y. Yao hysicists have long wondered whether and how iso- lated quantum systems thermalize—questions that are relevant to systems as diverse as ultracold atomic P gases and black holes. Recent theoretical and ex- perimental advances are bringing fresh insight into this line of inquiry. At one extreme, researchers have shown that disorder can fully arrest thermalization in certain isolated many-body quantum systems [1]. At the other extreme, Figure 1: A classical chaotic system can be diagnosed by the surprising results from the field of quantum gravity have presence of the buttery effect, in which a small perturbation like established that black holes are, in some sense, the fastest the tiny ap of a buttery's wing has a huge effect on the system at thermalizers in nature [2–4]. A common thread running some later point in time. (Left) Another version of the classical through these developments is an emerging focus on the dy- buttery effect compares the situations of running time forward (blue line) with running it backward after the buttery is still (white) namics of quantum information, in which thermalization is or after the buttery aps its wings (red). Without the buttery ap, associated with “scrambling,” or the loss of accessible infor- the system returns to its initial state; with it, the state of the system mation. Two groups, one in China [5] and one in the US [6], eventually differs drastically from its initial state. (Right) Li et al. [5] have taken a step towards tracking this scrambling of infor- and Gärttner et al. [6] performed an analogous experiment with mation in systems of quantum spins. quantum spin systems, here described by a wave function Y. Both groups used quantum-control techniques to evolve their systems The lore of thermalization goes as follows. Suppose you forward in time (blue line), to apply a perturbation W, and to evolve initialize a collection of quantum spins into one of two dis- the systems backward in time (red line). They then performed a tinct configurations. Now couple the system to a large heat measurement of V to diagnose the effect of the perturbation. bath. After equilibrium is reached, the final state of the spins (APS/Alan Stonebraker) will be independent of the spins’ initial configuration. In other words, information about the initial state of the spins has been irrevocably lost to the bath. radiation? Since a black hole is fundamentally a thermal ob- ject, this paradox is intimately related to how information But thermalization does not require a bath to proceed. In dynamics leads to thermalization. Specifically, one could a complex many-body quantum system, information about imagine that when something falls into a black hole, the in- the initial state may instead be “hidden” in elaborate corre- formation about it is encoded—albeit in scrambled form—in lations among the system’s constituents. The information in the radiation emitted during evaporation. such a scrambled state is not lost, because the final state can be related to the initial state by a unitary transformation. But Experiments that can probe the quantum dynamics of it may be inaccessible to any reasonable local measurement. black holes are currently out of reach. But scrambling is also relevant to isolated collections of strongly interacting atoms, The concept of information scrambling first arose in at- ions, molecules, and photons—all systems that physicists tempts to understand the black hole information paradox, can prepare in the lab. As a bonus, it may be possible to which asks: How can information about what fell into a engineer Hamiltonians in these systems that scramble infor- black hole be both trapped inside the event horizon and lib- mation as fast as black holes. The most direct way to detect erated as the black hole “evaporates” by emitting Hawking scrambling would be to measure a system’s entropy over time, though this is typically too hard to do. Instead, re- Department of Physics, University of Maryland, College Park, MD searchers have figured out that they can partially diagnose scrambling using unusual correlation functions called out- 20740, USA Department of Physics, University of California, Berkeley, CA of-time-order correlators (OTOCs) [2, 3, 7]. These correlators 94720, USA effectively involve a many-body “time machine.” Given two physics.aps.org 2017 American Physical Society 19 July 2017 Physics 10, 82 simple quantum operators W and V, one imagines compar- systems is well beyond what physicists can simulate on ing two processes: (i) Evolve the system forward in time, a classical computer. However, the team confined its ex- apply W, evolve backward in time, and apply V; (ii) apply periment to the dynamics of a more tractable subspace of V, evolve forward in time, apply W, and evolve backward quantum states. Moreover, despite the large number of spins in time. in their experiment, the spin Hamiltonian that they engi- neered was not chaotic, and their measurements of OTOCs, What does this comparison tell you? Drawing on an anal- like those of the molecular spin experiment, took place far ogy to classical chaos, one interpretation is that comparing from the limit of many-body scrambling. the two processes reveals the sensitivity of a measurement While neither experiment reaches the limit of true many- of V to a perturbation W—say, a kick from an external body chaos, both raise crucial questions. Can one dis- field—that happened some time in the past. If the mea- tinguish information that is scrambled from that which is surement is very sensitive to the perturbation, we have a simply lost because of environmental noise and spin deco- quantum version of the classical butterfly effect, in which herence? Can one correct for small errors that result from a small initial perturbation eventually has a major effect imperfectly evolving a system backward in time? Using the (Fig. 1). Taking the analogy further, a quantum system in rich data set from their ion experiment, Gärttner et al. were which information becomes scrambled can be viewed as a able to explore and model various sources of such imperfec- quantum chaotic system, and the OTOC provides a measure tion such as magnetic-field noise. of the scrambling. Quantum thermalization is a rapidly developing field. In Unfortunately, measuring OTOCs is difficult. Existing fact, two new scrambling experiments appeared just recently proposals [8–10] for doing so require the experimenter ei- [11, 12]. The near future promises experiments of increas- ther to evolve a system forward in time under a Hamiltonian ing complexity—both larger system sizes and more chaotic and then backwards in time by implementing the negative of Hamiltonians. Building on the work by Li et al. and Gärttner this Hamiltonian or to make a delicate comparison of two et al., it seems likely that experiments will soon forge beyond many-body quantum states. Thanks to the growing tool- what computers can simulate, revealing the dynamics of in- box of quantum-control techniques, these difficult tasks are formation scrambling in previously inaccessible regimes. now (somewhat) possible and the teams from China and the US have demonstrated proof-of-principle measurements of OTOCs. This research is published in Physical Review X and in Nature Jun Li, from the Beijing Computational Science Research Physics. Center, and colleagues used four nuclear spins in the iodotri- fluroethylene molecule [5]. After preparing the spins in a REFERENCES particular initial state, they applied a sequence of control pulses to engineer a quantum simulation of the mixed-field [1] R. Nandkishore and D. A. Huse, ``Many-Body Localization and Ising Hamiltonian, evolving this Hamiltonian forward in Thermalization in Quantum Statistical Mechanics,'' Annu. Rev. time. After perturbing the spins, they used another series of Cond. Matt. Phys. 6, 15 (2015). control pulses to implement the negative of the Ising Hamil- [2] S. H. Shenker and D. Stanford, ``Black Holes and the Buttery tonian, thus enabling the necessary “rewinding” of time, and Effect,'' J. High Energy Phys. 2014, 67 (2014). again evolved the spins. Their measurement of the final spin [3] A. Kitaev, Talk at Fundamental Physics Prize Symposium Nov. 10, 2014. state effectively yields the OTOC. But while their Hamilto- [4] J. Maldacena, S. H. Shenker, and D. Stanford, ``A Bound on nian is, in principle, chaotic, the system size is so small that Chaos,'' J. High Energy Phys. 2016, 106 (2016). its full evolution can be directly simulated on a computer, [5] J. Li, R. Fan, H. Wang, B. Ye, B. Zeng, H. Zhai, X. Peng, and it is far from the limit of many-body chaos. and J. Du, ``Measuring Out-of-Time-Order Correlators on a Nu- Martin Gärttner, from the University of Colorado Boulder, clear Magnetic Resonance Quantum Simulator,'' Phys. Rev. X JILA, and the National Institute of Standards and Technol- 7, 031011 (2017). ogy (all in Boulder, Colorado) [6], and colleagues studied [6] M. Gärttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. the dynamics of a much larger system consisting of more J. Bollinger, and A. M. Rey, ``Measuring Out-of-time-order 9 + Correlations and Multiple Quantum Spectra in a Trapped-ion than one hundred Be ions confined in a two-dimensional Quantum Magnet,'' Nat. Phys. (2017). electromagnetic trap. The valence electron spin of each ion [7] A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 55, behaves as an S = 1/2 magnetic moment. The Boulder 2262 (1969), [Sov. Phys. JETP 28, 1200 (1965)]. team implemented a long-range classical Ising Hamiltonian [8] B. Swingle, B. Bentsen, M. Schleier-Smith, and P. Hayden, by using a laser to couple the spins to the motional modes ``Measuring the Scrambling of Quantum Information,'' Phys. of the ion crystal. Then, using a protocol analogous to that Rev. A 94, 040302 (2016). of the team from China, they evolved the system “forward” [9] N. Y. Yao, F. Grusdt,, B. Swingle, M. D. Lukin, D. M. Stamper- and “backward” in time to measure the OTOC. The gen- Kurn, J. E. Moore, and E. A. Demler, ``Interferometric Approach eral dynamical evolution of one hundred two-level quantum to Probing Fast Scrambling,'' arXiv:1607.01801. physics.aps.org 2017 American Physical Society 19 July 2017 Physics 10, 82 [10] G. Zhu, M. Hafezi, and T. Grover, ``Measurement of Many-Body [12] E. J. Meier, J. Ang'ong'a, F. A. An, and B. Gadway, ``Exploring Chaos Using a Quantum Clock,'' Phys. Rev. A 94, 062329 Quantum Signatures of Chaos on a Floquet Synthetic Lattice,'' (2016). arXiv:1705.06714. [11] K. X. Wei, C. Ramanathan, and P. Cappellaro, ``Exploring Lo- calization in Nuclear Spin Chains,'' arXiv:1612.05249. 10.1103/Physics.10.82 physics.aps.org 2017 American Physical Society 19 July 2017 Physics 10, 82

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