# Quantum Thermodynamic Uncertainties in Nonequilibrium Systems from Robertson-Schrödinger Relations

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## Abstract

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## 1. Introduction

#### 1.1. Our Intents, Modeling and Methodology

#### 1.2. Our Findings in Relation to Background Works

- This work aims at exploring the relationship between thermodynamic and quantum uncertainty principles, as well as the existence and meaning of fluctuation-dissipation inequalities. Related to this work are two groups of our earlier papers:
- (a)
- The uncertainty principle at finite temperature has been shown (e.g., [7,9]) to be a useful indicator of quantum to classical transition, in that a crossover temperature can be identified between a vacuum fluctuations-dominated regime at very low temperatures to a thermal fluctuations-dominated high temperature regime where conventional thermodynamics applies.
- (b)
- A quantum fluctuation-dissipation inequality exists in a thermal quantum bath [19]: quantum fluctuations are bounded below by quantum dissipation, whereas classically the fluctuations vanish at zero temperature. The lower bound of this inequality is exactly satisfied by (zero-temperature) quantum noise and is in accord with the Heisenberg uncertainty principle. This inequality has been applied to understand issues in quantum friction [47] (see, e.g., [29] for background). A good summary of recent work on the relation of thermodynamic uncertainty relations and non-equilibrium fluctuations can be found in, e.g., [18] and references therein.

- Toward our stated goals, as a preamble, we can make this categorical statement: For Gaussian systems, thermodynamic functions are functionals of the Robertson-Schrödinger uncertainty (RSU) function (11), which is always non-negative for quantum systems, but not necessarily so for classical systems. Here, quantum refers to noncommutativity of the canonical operator pairs.
- The expectation value of the nonequilibrium Hamiltonian of mean force [48] gives the nonequilibrium internal energy of the system, and is bounded from above by the expectation value of the system’s Hamiltonian.
- The nonequilibrium heat capacity, derived from taking the derivative of the nonequilibrium internal energy with respect to the nonequilibrium effective temperature, remains proportional to the fluctuations of nonequilibrium Hamiltonian of mean force, with a proportionality constant given by the nonequilibrium effective temperature, not the bath temperature. These results apply for all times and at strong coupling.
- From the nonequilibrium free energy [48], we succeeded in deriving several inequalities ((25) and (27)) between certain thermodynamic quantities. They assume the same forms as those in conventional thermodynamics, but emphatically, these are nonequilibrium in nature and they hold for all times and at strong coupling.
- Fluctuation-dissipation inequalities (FDI) and relation with fluctuation-dissipation relation (FDR)
- (a)
- (b)
- We have shown that a fluctuation-dissipation inequality exists at all times in the nonequilibrium dynamics of the system.
- (c)
- At late times in the nonequilibrium relaxation of the reduced system, this fluctuation-dissipation inequality leads to the Robertson-Schrödinger uncertainty principle with the help of the Cauchy-Schwarz inequality.

- While the mathematical expressions of FDI have been found (6-b), we want to further understand their physical meanings. While the relation between FDI and RSU has been found (6-c) in the stage when the reduced system are closely approaching equilibrium, we want to find out whether there is a connection between the FDIs and the FDRs. This will provide a useful linkage between the more challenging nonequilibrium dynamics and the more familiar equilibrium states.

## 2. Nonequilibrium Dynamics of Gaussian Open Quantum Systems

#### 2.1. Dynamical Behavior of The Covariance Matrix

- Each element can be divided into an active and a passive component. The active component depends on the initial state of the system and represents the intrinsic quantum nature of the system. The passive component relies on the initial state of the environment and represents the induced quantum effects of the environment [26].
- Since ${d}_{1}\left(t\right)$, ${d}_{2}\left(t\right)$ decay with time, the active component will diminish at late times. The behavior of the covariance matrix elements at late times are essentially governed by the environment. In this way, the statistical and causality properties of the environment will eventually be passed on to the reduced system.
- The damping term in Equation (9) is proportional to the “velocity” of $\widehat{Q}$, i.e., the canonical momentum. Hence, if the reduced system has a small initial momentum uncertainty, then we expect that damping plays a minor role compared to the noise force at the early stage of the nonequilibrium evolution. Accordingly, the elements a, b will increase with time, mainly driven by the fluctuation force. In due course, the momentum uncertainty will grow sufficiently large such that the damping effect gradually catches up with the noise effect.
- In contrast, if the reduced system has a large initial momentum uncertainty, the damping effect will start off strongly, and the noise effect is subdominant. The elements a, b decrease with time until the damping effect is small enough to match up with the effect of the noise force.
- The element c does not have a definite sign, but oscillates with time. However, from parity consideration, when equilibrium is reached, the time- translational invariance of the state requires that c should vanish. Thus, $c=0$ may serve as an indicator of the existence of an asymptotic equilibrium state. In the final equilibrium state, both a and b are constant in time.
- The Hadamard function of the bath ${G}_{\mathrm{H}}^{\left(\varphi \right)}$ can be decomposed into two contributions in the current setting: One results from vacuum fluctuations of the field, the other from thermal fluctuations. We note that additional terms would appear if macroscopic bodies were present in the surroundings of the particle, which would carry their own (material-modified) quantum and thermal fluctuations [66]. At low bath temperatures ${\beta}_{\mathrm{B}}{\omega}_{\mathrm{R}}\gtrsim 1$, the vacuum fluctuation effects dominate, while at higher bath temperatures, they become insignificant. The vacuum fluctuations of the massless field are scaleless, such that the Hadamard function has a rather long range effect at the order of the squared inverse distance from the source. Again, if additionally material bodies were present, the situation can be different in the vicinity of surfaces where, depending on the structure of the material [67,68], often higher-order inverse polynonmials [61,69,70] or even non-algebraic functions [71,72] for the distance-dependence occur.
- The previous discussions and formalism are not restricted to weak coupling. They also apply to the case of strong coupling $\gamma /{\omega}_{\mathrm{R}}\sim \mathcal{O}\left(1\right)$, as long as the dynamics is stable. For strong coupling, since the scales of the reduced system like $\gamma $, ${\omega}_{\mathrm{R}}$, and the bath temperature ${\beta}_{\mathrm{B}}^{-1}$ can become comparable in magnitude, the curves for the temporal evolution of the covariance elements of the reduced system can show a rich structure during the nonequilibrium evolution (see, e.g., Figure 1).

#### 2.2. Robertson-Schrödinger Uncertainty Principle

## 3. Nonequilibrium Quantum Thermodynamics and Uncertainty Relations

#### 3.1. Nonequilibrium Partition Function and Effective Temperature

#### Effective Temperature in a Dynamical Setting

#### 3.2. Hamiltonian of Mean Force and Internal Energy

#### 3.2.1. Internal Energy

#### 3.2.2. Nonequilibrium Thermodynamic Inequalities

#### 3.2.3. Heat Capacity

## 4. Fluctuation-Dissipation Inequality and Robertson-Schrödinger Uncertainty

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Kramers-Kronig Relation for the Green’S Function

## Appendix B. The Robertson-Schrödinger Uncertainty Principle

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**Figure 1.**The time dependence of the uncertainty function $\mathfrak{S}\left(t\right)$ is plotted for three different bath temperatures. From the left to the right, the inverse bath temperature ${\beta}_{\mathrm{B}}$ is 10, 1, 0.1, normalized with respect to ${\omega}_{\mathrm{R}}^{-1}$. The damping constant $\gamma $ is $0.3\times {\omega}_{\mathrm{R}}$ and the oscillator mass m is $m=1\times {\omega}_{\mathrm{R}}$. In the low temperature regime ${\beta}_{\mathrm{B}}{\omega}_{\mathrm{R}}>1$, the finite temperature contribution is subdominant, and the effects due to vacuum fluctuations of the bath and its cutoff are more prominent.

**Figure 2.**The dependence of the uncertainty function $\sqrt{\mathfrak{S}\left(t\right)}$ at late times on the bath temperatures. We choose the time t to be $t=10{\omega}_{\mathrm{R}}^{-1}$ so that $\gamma t=3$, i.e., the regime where the relaxation is almost complete. We take the square root of the uncertainty function to reveal the linear trend at sufficiently high bath temperature where the equipartition theorem applies. We choose the same values for the other parameters used in Figure 1.

**Figure 3.**We show the time dependence of the effective temperature, and the bath temperature is drawn as the reference. The top row corresponds to the low bath temperature cases, and the bottom row to the high bath temperature. Here the temperature is normalized to the oscillator frequency. The three columns represent $\gamma =0.05$, $\gamma =0.3$ and $\gamma =0.6$ from left to right. The cutoff parameter $\u03f5$ is $\u03f5=0.05$. We observe that the gap between the effective temperature ${T}_{\mathrm{eff}}$ and the bath temperature ${T}_{\mathrm{B}}$ at late times decreases with weaker system-bath coupling. Thus they are best matched in the limits of high bath temperature and vanishing coupling strength, the common setting in the traditional thermodynamics.

**Figure 4.**We show the cutoff dependence of the effective temperature at late times, with the bath temperature as the reference. Its numerical value depends on the frequency cutoff and the regularization scheme. Here we choose ${T}_{\mathrm{B}}=10\phantom{\rule{0.166667em}{0ex}}{\omega}_{\mathrm{R}}$, and pick the soft cutoff scheme, that is, inserting a convergent factor of the form ${e}^{-\kappa \u03f5}$ in the frequency integral over $\kappa $. The cutoff frequency is of the order ${\u03f5}^{-1}$. The effective temperature can be smaller than the bath’s initial temperature if we choose a smaller cutoff frequency.

**Figure 5.**The time dependence of various candidates of internal energy. The top row corresponds to the case $\gamma =0.3$, and the bottom row for $\gamma =0.6$. Here the temperature is normalized to the oscillator frequency. The three columns represent ${\beta}_{\mathrm{B}}=10$, ${\beta}_{\mathrm{B}}=1$ and ${\beta}_{\mathrm{B}}=0.11$ from left to right, respectively corresponding to the high-, medium-, and low-temperature regime. Observe that $\langle {\widehat{H}}_{\mathrm{S}}\rangle $ is always greater or equal to ${\mathcal{U}}_{\mathrm{S}}=\langle {\widehat{H}}_{\mathrm{eff}}\rangle $ and the difference between them is smaller with higher bath temperature or weak system-bath coupling. By contrast, $\langle {\widehat{H}}_{\mathrm{MF}}\rangle $ is not grouped well together with the other two internal energies. This is related to the behavior of ${\beta}_{\mathrm{eff}}/{\beta}_{\mathrm{S}}$ shown in Figure 3.

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**MDPI and ACS Style**

Dong, H.; Reiche, D.; Hsiang, J.-T.; Hu, B.-L.
Quantum Thermodynamic Uncertainties in Nonequilibrium Systems from Robertson-Schrödinger Relations. *Entropy* **2022**, *24*, 870.
https://doi.org/10.3390/e24070870

**AMA Style**

Dong H, Reiche D, Hsiang J-T, Hu B-L.
Quantum Thermodynamic Uncertainties in Nonequilibrium Systems from Robertson-Schrödinger Relations. *Entropy*. 2022; 24(7):870.
https://doi.org/10.3390/e24070870

**Chicago/Turabian Style**

Dong, Hang, Daniel Reiche, Jen-Tsung Hsiang, and Bei-Lok Hu.
2022. "Quantum Thermodynamic Uncertainties in Nonequilibrium Systems from Robertson-Schrödinger Relations" *Entropy* 24, no. 7: 870.
https://doi.org/10.3390/e24070870