Single-Atom Demonstration of the Quantum Landauer Principle

One of the outstanding challenges to information processing is the eloquent suppression of energy consumption in the execution of logic operations. The Landauer principle sets an energy constraint in deletion of a classical bit of information. Although some attempts have been made to experimentally approach the fundamental limit restricted by this principle, exploring the Landauer principle in a purely quantum mechanical fashion is still an open question. Employing a trapped ultracold ion, we experimentally demonstrate a quantum version of the Landauer principle, i.e., an equality associated with the energy cost of information erasure in conjunction with the entropy change of the associated quantized environment. Our experimental investigation substantiates an intimate link between information thermodynamics and quantum candidate systems for information processing.

Figure 1

A single trapped ion manipulated for exploring an improved LP. (a) Level scheme of the ${{}^{40}\text{Ca}}^{+}$ ion confined in a linear Paul trap under an external magnetic field. The qubit encoded in $|\uparrow ⟩$ and $|\downarrow ⟩$ represents the system, and the reservoir is denoted by the quantized $z$-axis vibration of the ion. The system-reservoir coupling is well controlled by red- or blue-sideband 729-nm laser radiation, and readout of the qubit is carried out by 397-nm spontaneous emission. (b) Illustrative schematic. (1) Starting point, where the qubit (i.e., the system) is in a maximally mixed state ${\rho }_S=(|\uparrow ⟩⟨\uparrow |+|\downarrow ⟩⟨\downarrow |)/2$ and the vibrational degree of freedom (i.e., the reservoir) is thermalized to energy $E_0$. (2) Intermediate step, where the 729-nm laser drives a unitary evolution of the system and reservoir. The entropy of the system decreases, while the energy of the reservoir increases to $E_m$. (3) End of the erasure, where the system is populated in $|\downarrow ⟩$ and the reservoir ends at a higher level with energy $E_f$. (c) Flow diagram of the experimental operations, where (1) is to reach a mixed state of the system, (2) is related to a thermal state of the reservoir, (3) is to monitor the population of the system state, and (4) refers to measuring the average phonon number in the reservoir by observing the blue-sideband Rabi oscillation.

Figure 2

Experimental implementation of information erasure in a qubit system. (a) Blue-sideband Rabi oscillations before the erasure for different waiting time $\tau$. (b) Blue-sideband Rabi oscillation at the end of erasure for different waiting time $\tau$. (c) Average phonon number in the reservoir before the erasure (red dots) and after the erasure (black circles), obtained by calculating the data in (a) and (b). (d) Population in $|\downarrow ⟩$ (black circles) and $|\uparrow ⟩$ (red dots) at the end of erasure. The system is initialized with populations of 0.531(15) and 0.467(16) in $|\downarrow ⟩$ and $|\uparrow ⟩$, respectively. Dots or circles are experimental data and curves are analytical results. Each data point in (a) and (b) is measured by $10^2$ repetition and in (d) by $10^4$ repetition. (e) Test of the improved LP at different reservoir temperatures. The curves from top to bottom correspond to $\mathrm{\Delta }Q/k_{B}T$ with $\mathrm{\Delta }Q=Q_0(⟨n⟩-⟨n{⟩}_0)$ (black curve and crosses), $\mathrm{\Delta }S+I(S^{\prime }:R^{\prime })+D({\rho }_R^{\prime }\parallel {\rho }_R)$ (red curve and circles), and $\mathrm{\Delta }S$ (blue curve and dots), where the reservoir temperature is defined as $T=T_0/\ln [1+1/⟨n{⟩}_0]$, with $⟨n{⟩}_0$ the initial average phonon number. Inset of (e) demonstrates the result of $\mathrm{\Delta }Q/k_{B}T-[\mathrm{\Delta }S+I(S^{\prime }:R^{\prime })+D({\rho }_R^{\prime }\parallel {\rho }_R)]$. Experimental data in (e) are obtained from (a)–(d). The energy and temperature units are $Q_0≔\hslash {\omega }_z=6.7\times {10}^{-28}\mathrm{J}$ and $T_0≔Q_0/k_B=48.5\mu \mathrm{K}$, respectively. The curves in each panel are analytical results in our optimal erasure scheme using a $\pi$ pulse of the red-sideband radiation [34]. The associated error bars reveal standard deviation.

Figure 3

Experimental test of the improved LP under different initial states of the system. Curves from top to bottom correspond to $\mathrm{\Delta }Q/k_{B}T$ (black curve and crosses), $\mathrm{\Delta }S+I(S^{\prime }:R^{\prime })+D({\rho }_R^{\prime }\parallel {\rho }_R)$ (red curve and circles), and $\mathrm{\Delta }S$ (blue curve and dots), where the reservoir temperature is initially $18.2(8)\mu \mathrm{K}$ corresponding to the average phonon number $⟨n{⟩}_0=0.074(9)$. The horizontal dashed line indicates the positive-negative switch of the entropy decrease with boundary points at ${\theta }_c=0.54(3)$ and 2.80(2). Inset demonstrates the result of $\mathrm{\Delta }Q/k_{B}T-[\mathrm{\Delta }S+I(S^{\prime }:R^{\prime })+D({\rho }_R^{\prime }\parallel {\rho }_R)]$. Dots, circles, and crosses represent experimental data; curves are from analytical results. The system entropy is initially $S_1=S\{\mathrm{diag}[{\cos }^2({\theta }_c/2),{\sin }^2({\theta }_c/2)]\}$, with diag indicating a diagonal matrix and ${\theta }_c$ associated with the initial state of the system. The error bars indicate standard deviation.