Experimental test of Landauer’s principle in single-bit operations on nanomagnetic memory bits

Hong, Jeongmin; Lambson, Brian; Dhuey, Scott; Bokor, Jeffrey

doi:10.1126/sciadv.1501492

Cited by 179 publications

(179 citation statements)

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“…This limit has been verified in a variety of classical systems, including particles in traps 2,3 and nanomagnets 4 . Here, we extend it to the quantum realm by using a crystal of molecular nanomagnets as a quantum spin memory and showing that its erasure is still governed by the Landauer principle.…”

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confidence: 97%

“…This has successfully produced stateof-the-art systems with picosecond timescales, although operating far (≳ 10 6 ) above the reversible limit [9][10][11][12] . On the other hand, reducing W down to the Landauer limit, at the expense of slow operation, has been beautifully demonstrated using small particles in traps 2,3 or single-domain nanomagnets 4 as envisioned by Landauer and Bennett 1,8 . All of the mentioned systems are large enough to be treated as classical.…”

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confidence: 99%

“…χ z is proportional to the derivative of the magnetization ∂M/∂H and is a function of the temperature T, the frequency ω/2π of the a.c.-field H a.c. and the magnetic field vector H. The magnetization and work can be derived by integrating χ ′ z once and twice with respect to the d.c. magnetic field, respectively. The work W obtained in this manner quantifies the entropy produced during the erasure and measures how reversible the reset operation is 4,6 .…”

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Quantum Landauer erasure with a molecular nanomagnet

et al. 2018

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The erasure of a bit of information is an irreversible operation whose minimal entropy production of k B ln 2 is set by the Landauer limit 1 . This limit has been verified in a variety of classical systems, including particles in traps 2,3 and nanomagnets 4 . Here, we extend it to the quantum realm by using a crystal of molecular nanomagnets as a quantum spin memory and showing that its erasure is still governed by the Landauer principle. In contrast to classical systems, maximal energy efficiency is achieved while preserving fast operation owing to its high-speed spin dynamics. The performance of our spin register in terms of energy-time cost is orders of magnitude better than existing memory devices to date. The result shows that thermodynamics sets a limit on the energy cost of certain quantum operations and illustrates a way to enhance classical computations by using a quantum system.While a computation performed with an ideal binary logic gate (for example, NOT) has no lower energy dissipation limit [5][6][7] , one carried out in a memory device does. The reason is that in the former the bit is merely displaced isentropically in the space of states, whereas in the latter the minimal operation, called Landauer erasure, entails resetting the memory irrespective of its initial state. Let us see how this erasure applies to a classical N-bit register (Fig. 1a, left) and how the Landauer limit comes about. In the first stage, each bit of the register, in a definite state '0' or '1' , is allowed to explore the two binary states by lowering the potential barrier and through the action of temperature fluctuations. This doubling of the phase space is accompanied by an entropy production Δ S = k B ln 2 per bit, where k B is the Boltzmann constant. In the second stage, a work W ≥ TΔ S is required to reduce the register's entropy and phase space to their initial values 8 . The limit W = TΔ S is reached only if this reduction is carried out reversibly. This can be achieved when using a frictionless system in a quasi-static fashion (that is, at timescales slower than its relaxation time τ rel ), so that unwanted memory and hysteresis effects are avoided. For this reason, slow (fast) operation-with respect to the system-dependent τ rel -is generally associated with a lower (higher) dissipation.This complementarity between work and time suggests considering the product Wτ rel -rather than either of the two-as the figure of merit assessing the energy-time cost of a computation. On one hand, driven by the demand for speed, effort has been put into pursuing fast-switching storage devices. This has successfully produced stateof-the-art systems with picosecond timescales, although operating far (≳ 10 6 ) above the reversible limit [9][10][11][12] . On the other hand, reducing W down to the Landauer limit, at the expense of slow operation, has been beautifully demonstrated using small particles in traps 2,3 or single-domain nanomagnets 4 as envisioned by Landauer and Bennett 1,8 . All of the mentioned systems are large enough to ...

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Quantum Landauer erasure with a molecular nanomagnet

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“…Finally, under these conditions, the change in non-equilibrium free energy is simply, from Eq. 10,This is the principle proposed by Landauer in 1961 [35] and studied extensively since [41,42,44,[46][47][48][49][50][51][52][53][54][55][56][57][58][59], with recent experimental confirmation [60][61][62][63][64]. Thus, by measuring the minimal average work to carry out protocols that alter the information content of a two-state system, we can test whether the Shannon entropy has physical relevance: Does it apply to thermodynamic descriptions such as Eq.…”

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Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

Gavrilov

Chétrite

Bechhoefer

2017

Proc. Natl. Acad. Sci. U.S.A.

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Stochastic thermodynamics extends classical thermodynamics to small systems in contact with one or more heat baths. It can account for the effects of thermal fluctuations and describe systems far from thermodynamic equilibrium. A basic assumption is that the expression for Shannon entropy is the appropriate description for the entropy of a nonequilibrium system in such a setting. Here, for the first time, we measure experimentally this function. Our system is a micron-scale colloidal particle in water, in a virtual double-well potential created by a feedback trap. We measure the work to erase a fraction of a bit of information and show that it is bounded by the Shannon entropy for a twostate system. Further, by measuring directly the reversibility of slow protocols, we can distinguish unambiguously between protocols that can and cannot reach the expected thermodynamic bounds. INTRODUCTIONBeginning with the foundational work of Clausius, Maxwell, and Boltzmann in the 19th c., the concept of entropy has played a key role in thermodynamics. Yet, despite its importance, entropy is an elusive concept [1][2][3][4][5][6][7][8], with no unique definition; rather, the appropriate definition of entropy depends on the scale, relevant thermodynamic variables, and nature of the system, with ongoing debate existing over the proper definition even for equilibrium cases [9]. Moreover, entropy has not been directly measured but is rather inferred from other quantities, such as the integral of the specific heat divided by temperature. Here, by measuring the work required to erase a fraction of a bit of information, we isolate directly the change in entropy in an open nonequilibrium system, showing that it is compatible with the functional form proposed by Gibbs and Shannon, giving it a physical meaning in this context. Knowing the relevant form of entropy is crucial for efforts to extend thermodynamics to systems out of equilibrium.For a continuous classical system whose state in phase space x is distributed as the probability density function ρ(x), the Gibbs-Shannon entropy is [10,11] where k B is Boltzmann's constant. For quantum systems, von Neumann introduced, in 1927, the corresponding expression in terms of the density matrix [12]. Historically, the system in Eq. 1 has typically been assumed to be in thermal equilibrium. * email: johnb@sfu.caThe physical relevance of Eq. 1 for a nonequilibrium distribution ρ(x) has often been questioned (e.g., [13][14][15][16][17]). One concern is that S is constant on an isolated Hamiltonian system and can change only when evaluated on subsystems, such as those picked out by coarse graining. With many ways to choose subsystems or to coarse grain, is the associated notion of irreversibility intrinsic to the description of the system?In another approach to entropy, advanced in the context of communication and information theory, Shannon [11,18] proved that, up to a multiplicative constant, S is the only possible function satisfying three intuitive axioms. Alternatively, one can start from an...

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“…[9,10] used an anti-Brownian electrokinetic feedback trap and Hong et.al. [11] used nano magnetic memory bits to analyze the Landauer's bound. Studies in [8][9][10][11] demonstrate mechanisms for realizing memory and its erasure with energetics approaching Landauer's bound.…”

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Memory erasure using time-multiplexed potentials

2017

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We study the thermodynamics of a Brownian particle under the influence of a time multiplexed harmonic potential of finite width. The memory storage mechanism and the erasure protocol realized by time multiplexed potentials are utilized to experimentally realize erasure with work close to the Landauer's bound. We quantify the work done on the system with respect to the duty-ratio of time multiplexing, which also provides a handle to approach reversible erasures. A Langevin dynamics based simulation model is developed for the proposed memory bit and the erasure protocol, which guides the experimental realization. The study also provides insights into transport at the micro scale. PACS numbers: 05.40.-a, 05.70.LnThe Landauer's principle, pioneered by Rolf Landauer in 1961, provides a critical link between information theory and thermodynamics of physical systems [1]. It states that there is no process where the work done to erase one bit of information is less than k b T ln 2 (Landauer's bound), when the prior probability of the bit being in any of the two states is equal [2]. Here, k b is the Boltzmann constant and T is the temperature of the heat bath.Numerous analyses have established the Landauer's bound through different approaches [3][4][5][6][7]. The experimental study of Landauer's bound has only recently be- In this article, we study the stochastic energetics of transport realized by time multiplexing of a finite width harmonic potential to effectively realize a bi-stable potential. Here a single laser in an optical tweezer setup, is multiplexed at two locations with varying dwell times to create a series of potentials (symmetric as well as asymmetric bi-stable potentials), that effectuates the erasure process. Moreover, experimental variables to realize reversible erasure are identified and utilized for approaching the Landauer's bound. A Langevin framework for a Brownian particle under the influence of a time multiplexed laser is developed and is shown to obey quantitative trends observed in experiments. We use our method of shaping the potential, by changing the dwell time of multiplexing of the laser, to erase one bit of information. The ease of implementation and the high-resolution accounting of energetics enabled by photodiode based measurements are other advantages of the method reported. We resort to Sekimoto's stochastic energetics [12][13][14][15] framework to quantify the work done on the system in the erasure process. The underlying principles developed in this article are applicable toward the study of transport achieved by time multiplexing of a single potential, and are not restricted to realizations based on optical traps.Experiment and Simulation Model for one-bit Memory: A Brownian particle in a double well potential is used to model a one-bit memory. The memory is designated the state 'zero' if the particle is in the left well, and the state 'one' if it is in the right well. We realize a Brownian particle in a harmonic potential of finite width experimentally, by using a custom built op...

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Experimental test of Landauer’s principle in single-bit operations on nanomagnetic memory bits

Abstract: The minimum energy dissipated in switching a magnetic bit measured to be consistent with the Landauer limit of kBT ln(2).

Cited by 179 publications

References 22 publications

Quantum Landauer erasure with a molecular nanomagnet

Quantum Landauer erasure with a molecular nanomagnet

Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

Memory erasure using time-multiplexed potentials

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