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Heat production and error probability relation in Landauer reset at effective temperature

View Article: PubMed Central - PubMed

ABSTRACT

The erasure of a classical bit of information is a dissipative process. The minimum heat produced during this operation has been theorized by Rolf Landauer in 1961 to be equal to kBT ln2 and takes the name of Landauer limit, Landauer reset or Landauer principle. Despite its fundamental importance, the Landauer limit remained untested experimentally for more than fifty years until recently when it has been tested using colloidal particles and magnetic dots. Experimental measurements on different devices, like micro-mechanical systems or nano-electronic devices are still missing. Here we show the results obtained in performing the Landauer reset operation in a micro-mechanical system, operated at an effective temperature. The measured heat exchange is in accordance with the theory reaching values close to the expected limit. The data obtained for the heat production is then correlated to the probability of error in accomplishing the reset operation.

No MeSH data available.


Related in: MedlinePlus

Schematic of the whole system and measurement setup.Lateral view of the whole system and measurement setup. Two magnets with opposite magnetic orientations are used to induce bistability in the system. Two electrodes are used to apply electrostatic forces on the mechanical structure: VL and VR to force the cantilever to bend to the left (negative x) and to the right (positive x) respectively. The magnetic interaction can be engineered by changing geometric parameters such as d and Δx. (b) Color-map of the reconstructed potential energy as function of the distance between the magnets, d. The distance is expressed in arbitrary units proportional to the voltage applied to the piezoelectric stage. Decreasing the distance between the magnets the potential energy softens and eventually two stable states appear. (c) Dependence of the effective temperature, Teff, with the root mean square of the white Gaussian voltage applied to the piezoelectric shaker. The red dot represents the condition accounted for the experimental data presented.
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f1: Schematic of the whole system and measurement setup.Lateral view of the whole system and measurement setup. Two magnets with opposite magnetic orientations are used to induce bistability in the system. Two electrodes are used to apply electrostatic forces on the mechanical structure: VL and VR to force the cantilever to bend to the left (negative x) and to the right (positive x) respectively. The magnetic interaction can be engineered by changing geometric parameters such as d and Δx. (b) Color-map of the reconstructed potential energy as function of the distance between the magnets, d. The distance is expressed in arbitrary units proportional to the voltage applied to the piezoelectric stage. Decreasing the distance between the magnets the potential energy softens and eventually two stable states appear. (c) Dependence of the effective temperature, Teff, with the root mean square of the white Gaussian voltage applied to the piezoelectric shaker. The red dot represents the condition accounted for the experimental data presented.

Mentions: The mechanical system used to perform the experiment is depicted in Fig. 1(a). A triangular micro-cantilever, 200 μm long, is used to encode one bit on information. In order to obtain two stable states two magnets with opposite magnetization are placed on the tip of the cantilever and on a movable stage facing the cantilever. In this way, depending on the distance between the magnets, d, and the relative lateral alignment, Δx, it is possible to induce bistability on the system. Figure 1(b) shows the potential energy as a function of d reconstructed from the probability density function of the position of the cantilever at equilibrium, ρ(x, d) = Aexp(−U(x, d)/kBT), which implies that U(x, d) = −kBT lnρ(x, d) + U02. When the magnets are far away the effect of the repulsive force is negligible, the system is then monostable and can be approximated to a linear system. Decreasing the distance the repulsive force between magnets tends to soften the system up to the point where two stable positions appear. The effect of reducing even more d is to enlarge the separation of the rest states and to increase the potential barrier separating these two wells. Eventually, when the distance between the magnets is small enough, the system remains trapped in one well for a period of time larger than the relaxation time of the system. Logic states are encoded in the position of the cantilever tip: logic 0 for x < 0 and logic 1 for x > 0. The proposed system presents intrinsic dissipative processes that depend on the maximum displacement of the cantilever tip6. The minimum heat produced when performing a physical transformation of the system is proportional to . In our setup it is not possible to reduce the separation between the two potential wells to a value that bounds the heat produced by intrinsic dissipation below the Landauer limit. Increasing the effective temperature increases the value for the Landauer limit making possible to have negligible intrinsic dissipation. A piezoelectric shaker is used to excite the structure with a band limited white Gaussian noise to mimic the effect of an arbitrary temperature. In the present experiment the white noise is limited to 50 kHz, well above the resonance frequency of the free cantilever (f0 = 5.3 kHz). The dependence of the effective temperature with the root-mean-squared voltage supplied to the shaker is reported in Fig. 1(c). The red dot, corresponding to an effective temperature of Teff = 5 × 107 K, highlights the condition considered in the present case. The solid line represent the expected trend where T ∝ 8. The effective temperature has been estimated computing the power spectral density (PSD) of the system at various piezoelectric noise excitation voltages. The obtained curves have been fitted with Lorenzian curves taking as reference for the calibration the one at room temperature (T = 300 K). The other curves have been used to extract the only varying parameter Teff, corresponding to the effective temperature of the system under external excitation. Finally two electrostatic probes, placed one on the left and the other on the right of the cantilever, are used to apply a negative and positive forces respectively. When a voltage different to zero is applied on one probe the cantilever feels an attractive electrostatic force toward the probe due to the polarization of the cantilever itself. The voltage on the probes, the distance between the magnets and their time evolution are used to specify the protocols used in order to change the bit stored in the system as described in the following subsection.


Heat production and error probability relation in Landauer reset at effective temperature
Schematic of the whole system and measurement setup.Lateral view of the whole system and measurement setup. Two magnets with opposite magnetic orientations are used to induce bistability in the system. Two electrodes are used to apply electrostatic forces on the mechanical structure: VL and VR to force the cantilever to bend to the left (negative x) and to the right (positive x) respectively. The magnetic interaction can be engineered by changing geometric parameters such as d and Δx. (b) Color-map of the reconstructed potential energy as function of the distance between the magnets, d. The distance is expressed in arbitrary units proportional to the voltage applied to the piezoelectric stage. Decreasing the distance between the magnets the potential energy softens and eventually two stable states appear. (c) Dependence of the effective temperature, Teff, with the root mean square of the white Gaussian voltage applied to the piezoelectric shaker. The red dot represents the condition accounted for the experimental data presented.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
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getmorefigures.php?uid=PMC5037424&req=5

f1: Schematic of the whole system and measurement setup.Lateral view of the whole system and measurement setup. Two magnets with opposite magnetic orientations are used to induce bistability in the system. Two electrodes are used to apply electrostatic forces on the mechanical structure: VL and VR to force the cantilever to bend to the left (negative x) and to the right (positive x) respectively. The magnetic interaction can be engineered by changing geometric parameters such as d and Δx. (b) Color-map of the reconstructed potential energy as function of the distance between the magnets, d. The distance is expressed in arbitrary units proportional to the voltage applied to the piezoelectric stage. Decreasing the distance between the magnets the potential energy softens and eventually two stable states appear. (c) Dependence of the effective temperature, Teff, with the root mean square of the white Gaussian voltage applied to the piezoelectric shaker. The red dot represents the condition accounted for the experimental data presented.
Mentions: The mechanical system used to perform the experiment is depicted in Fig. 1(a). A triangular micro-cantilever, 200 μm long, is used to encode one bit on information. In order to obtain two stable states two magnets with opposite magnetization are placed on the tip of the cantilever and on a movable stage facing the cantilever. In this way, depending on the distance between the magnets, d, and the relative lateral alignment, Δx, it is possible to induce bistability on the system. Figure 1(b) shows the potential energy as a function of d reconstructed from the probability density function of the position of the cantilever at equilibrium, ρ(x, d) = Aexp(−U(x, d)/kBT), which implies that U(x, d) = −kBT lnρ(x, d) + U02. When the magnets are far away the effect of the repulsive force is negligible, the system is then monostable and can be approximated to a linear system. Decreasing the distance the repulsive force between magnets tends to soften the system up to the point where two stable positions appear. The effect of reducing even more d is to enlarge the separation of the rest states and to increase the potential barrier separating these two wells. Eventually, when the distance between the magnets is small enough, the system remains trapped in one well for a period of time larger than the relaxation time of the system. Logic states are encoded in the position of the cantilever tip: logic 0 for x < 0 and logic 1 for x > 0. The proposed system presents intrinsic dissipative processes that depend on the maximum displacement of the cantilever tip6. The minimum heat produced when performing a physical transformation of the system is proportional to . In our setup it is not possible to reduce the separation between the two potential wells to a value that bounds the heat produced by intrinsic dissipation below the Landauer limit. Increasing the effective temperature increases the value for the Landauer limit making possible to have negligible intrinsic dissipation. A piezoelectric shaker is used to excite the structure with a band limited white Gaussian noise to mimic the effect of an arbitrary temperature. In the present experiment the white noise is limited to 50 kHz, well above the resonance frequency of the free cantilever (f0 = 5.3 kHz). The dependence of the effective temperature with the root-mean-squared voltage supplied to the shaker is reported in Fig. 1(c). The red dot, corresponding to an effective temperature of Teff = 5 × 107 K, highlights the condition considered in the present case. The solid line represent the expected trend where T ∝ 8. The effective temperature has been estimated computing the power spectral density (PSD) of the system at various piezoelectric noise excitation voltages. The obtained curves have been fitted with Lorenzian curves taking as reference for the calibration the one at room temperature (T = 300 K). The other curves have been used to extract the only varying parameter Teff, corresponding to the effective temperature of the system under external excitation. Finally two electrostatic probes, placed one on the left and the other on the right of the cantilever, are used to apply a negative and positive forces respectively. When a voltage different to zero is applied on one probe the cantilever feels an attractive electrostatic force toward the probe due to the polarization of the cantilever itself. The voltage on the probes, the distance between the magnets and their time evolution are used to specify the protocols used in order to change the bit stored in the system as described in the following subsection.

View Article: PubMed Central - PubMed

ABSTRACT

The erasure of a classical bit of information is a dissipative process. The minimum heat produced during this operation has been theorized by Rolf Landauer in 1961 to be equal to kBT ln2 and takes the name of Landauer limit, Landauer reset or Landauer principle. Despite its fundamental importance, the Landauer limit remained untested experimentally for more than fifty years until recently when it has been tested using colloidal particles and magnetic dots. Experimental measurements on different devices, like micro-mechanical systems or nano-electronic devices are still missing. Here we show the results obtained in performing the Landauer reset operation in a micro-mechanical system, operated at an effective temperature. The measured heat exchange is in accordance with the theory reaching values close to the expected limit. The data obtained for the heat production is then correlated to the probability of error in accomplishing the reset operation.

No MeSH data available.


Related in: MedlinePlus