Correlated vs. uncorrelated noise acting on a quantum refrigerator

TL;DR

This paper investigates how correlated and uncorrelated noise sources influence the dynamics of a two-qubit quantum refrigerator, revealing that noise correlation affects transition rules, system states, and cooling efficiency.

Contribution

It demonstrates that noise correlation alters transition selection rules and system dynamics, impacting the performance of a quantum refrigerator.

Findings

Fully correlated or anticorrelated noise creates protected states.

System dynamics depend on initial states under correlated noise.

Cooling power does not scale with qubit number under strong noise correlation.

Abstract

Two qubits form a quantum four-level system. The golden-rule based transition rates between these states are determined by the coupling of the qubits to noise sources. We demonstrate that depending on whether the noise acting on the two qubits is correlated or not, these transitions are governed by different selection rules. In particular, we find that for fully correlated or anticorrelated noise, there is a protected state, and the dynamics of the system depends then on its initialization. For nearly (anti)correlated noise, there is a long time scale determining the temporal evolution of the qubits. We apply our results to a quantum Otto refrigerator based on two qubits coupled to hot and cold baths. Even in the case when the two qubits do not interact with each other, the cooling power of the refrigerator does not scale with the number ($=2$ here) of the qubits when there is strong correlation of noise acting on them.

Equations

$$H=H_{Q1}+H_{Q2}+H_{\mathrm{S}}+H_{\mathrm{cS},1}+H_{\mathrm{cS},2},$$

$$H_{\mathrm{Q}j}=-E_j({\Delta }_j{\sigma }_{x,j}+q{\sigma }_{z,j}),$$

$${\lambda }_1=-2\sqrt{q^2+{\Delta }^2},{\lambda }_2={\lambda }_3=0,{\lambda }_4=+2\sqrt{q^2+{\Delta }^2}$$

$$|1⟩=\frac{1}{\sqrt{2}}(|u_1⟩+\frac{q}{\sqrt{q^2+{\Delta }^2}}|u_2⟩+\frac{\Delta }{\sqrt{q^2+{\Delta }^2}}|u_3⟩)$$

$$|2⟩=|u_4⟩$$

$$|3⟩=\frac{\Delta }{\sqrt{q^2+{\Delta }^2}}|u_2⟩-\frac{q}{\sqrt{q^2+{\Delta }^2}}|u_3⟩$$

$$|4⟩=\frac{1}{\sqrt{2}}(|u_1⟩-\frac{q}{\sqrt{q^2+{\Delta }^2}}|u_2⟩-\frac{\Delta }{\sqrt{q^2+{\Delta }^2}}|u_3⟩).$$

$$H_{\mathrm{cS}}\equiv \sum _{m=1,2}{H}_{\mathrm{cS},m}=\sum _{m=1,2}{\hat{A}}_m\delta {\hat{X}}_m(t),$$

$$⟨\delta {\hat{X}}_m(t^{\prime })\delta {\hat{X}}_n(t)⟩={\chi }_{mn}{\int }_{-\infty }^{\infty }\frac{d\omega }{2\pi }{e}^{-i\omega (t^{\prime }-t)}S(\omega ),$$

$${\Gamma }_{k\to l,\mathrm{S}}=\frac{1}{{\hslash }^2}\sum _{m,n=1}^2⟨k|{\hat{A}}_m|l⟩⟨l|{\hat{A}}_n|k⟩{\chi }_{mn}S({\omega }_{kl}),$$

$$\displaystyle{\bf\Gamma}=\left(\begin{array}[]{cccc}-2\Gamma_{\uparrow}&(1-\chi)\Gamma_{\uparrow}&(1+\chi)\Gamma_{\uparrow}&0\\ (1-\chi)\Gamma_{\downarrow}&-\Gamma_{\Sigma}(1-\chi)&0&(1-\chi)\Gamma_{\uparrow}\\ (1+\chi)\Gamma_{\downarrow}&0&-\Gamma_{\Sigma}(1+\chi)&(1+\chi)\Gamma_{\uparrow}\\ 0&(1-\chi)\Gamma_{\downarrow}&(1+\chi)\Gamma_{\downarrow}&-2\Gamma_{\downarrow}\end{array}\right)$$

$${\dot{\rho }}_{\mathrm{tot}}=\frac{i}{\hslash }[{\rho }_{\mathrm{tot}},H_{\mathrm{D},\mathrm{I}}(t)+H_{\mathrm{cS},I}(t)].$$

$$\displaystyle D=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cccc}1&q/\sqrt{q^{2}+\Delta^{2}}&\Delta/\sqrt{q^{2}+\Delta^{2}}&0\\ 0&0&0&\sqrt{2}\\ 0&\Delta/\sqrt{q^{2}+\Delta^{2}}&-q/\sqrt{q^{2}+\Delta^{2}}&0\\ 1&-q/\sqrt{q^{2}+\Delta^{2}}&-\Delta/\sqrt{q^{2}+\Delta^{2}}&0\end{array}\right).$$

$$\displaystyle\dot{\rho}_{kl}=\sum_{i=1}^{4}\big{\{}\rho_{ki}\langle i|D^{\dagger}\dot{D}|l\rangle e^{i{\Omega}^{-1}\int_{0}^{u}\lambda_{il}(u^{\prime})du^{\prime}}+\rho_{il}\langle i|D^{\dagger}\dot{D}|k\rangle e^{i{\Omega}^{-1}\int_{0}^{u}\lambda_{ki}(u^{\prime})du^{\prime}}+\delta_{kl}\rho_{ii}\Gamma_{i\rightarrow k}-\frac{1}{2}\rho_{kl}(\Gamma_{l\rightarrow i}+\Gamma_{k\rightarrow i})\big{\}}.$$

$$P_{\mathrm{B}}=\sum _{k,l}{\rho }_{kk}{E}_{kl}{\Gamma }_{k\to l,\mathrm{B}}.$$

$${\dot{\rho }}_d={\mathbf{\Gamma }}^{\mathrm{T}}{\rho }_d,$$

$${\rho }_{11}=\frac{1}{(1+r{)}^2},{\rho }_{22}={\rho }_{33}=\frac{r}{(1+r{)}^2},{\rho }_{44}=\frac{r^2}{(1+r{)}^2},$$

$$\displaystyle P_{\rm C}=\big{[}-(\Gamma_{1\rightarrow 2,{\rm C}}+\Gamma_{1\rightarrow 3,{\rm C}})\rho_{11}+(\Gamma_{2\rightarrow 1,{\rm C}}-\Gamma_{2\rightarrow 4,{\rm C}})\rho_{22}$$

$$\displaystyle+(\Gamma_{3\rightarrow 1,{\rm C}}-\Gamma_{3\rightarrow 4,{\rm C}})\rho_{33}+(\Gamma_{4\rightarrow 2,{\rm C}}+\Gamma_{4\rightarrow 3,{\rm C}})\rho_{44}\big{]}E$$

$$=2(-{\rho }_{gg}{\Gamma }_{\uparrow ,\mathrm{C}}+{\rho }_{ee}{\Gamma }_{\downarrow ,\mathrm{C}})E=2P_0,$$

Full text

Controlling the susceptibility of qubits to decoherence sources is a central issue in developing a robust quantum computer clarke ; devoret ; nielsen ; governale ; storcz ; pashkin ; averin ; you ; hu . Over the past two decades it has become obvious that the influence of a common source of noise on all qubits deviates dramatically from the situation where uncorrelated noise sources affect each individual qubit separately zanardi ; d'arrigo ; gustavsson ; galperin . In the first case, so called decoherence-free subspaces emerge, meaning that there are states that are not affected by the noise source. To realize robust quantum circuits and to provide error correction barends for them depends then on whether the noise sources are correlated or not averin1 ; clemens ; klesse ; novais .

In this Letter we demonstrate selection rules that account for the transitions in a two-qubit system affected by either a common noise source or multiple sources shown schematically for two extreme cases in Fig. 1. The basic four-level system of the two qubits exhibits then a protected state (”decoherence-free subspace”) when subjected to fully (anti)correlated noise, whereas for uncorrelated noise, all the four states couple to the noise. In our work we focus, instead of qubit decoherence, on energy transport between the baths at different temperatures producing the noise on the qubits. We study the dependence of this transmitted power on the initialization of the system and on the degree of noise correlation. To understand the influence of noise correlation in a physical system, we investigate a quantum Otto refrigerator Jukka ; niskanen2007 , a representative of quantum machines that are currently of considerable interest due to their experimental feasibility kay ; campisi2016 ; hofer2016 ; quan2007 ; rossnagel2016 ; Uzdin/Kosloff ; brandner2016 ; niskanen2007 ; Jukka .

The total Hamiltonian describing the system and the environment is given by

[TABLE]

where $H_{\rm Q1},H_{\rm Q2}$ are the Hamiltonians of the two (driven) qubits, $H_{\rm S}$ is the Hamiltonian of the noise source(s), and $H_{\rm cS,1},H_{\rm cS,2}$ the couplings of qubits 1 and 2 to the noise source(s). For our main argument we may assume that the two qubits are mutually decoupled although the selection rules to be presented hold also for coupled qubits. In the quantitative analysis, we use the four Bell basis states $\{|u_{1}\rangle=\frac{1}{\sqrt{2}}(|0_{1}0_{2}\rangle+|1_{1}1_{2}\rangle),~{}|u_{2}\rangle=\frac{1}{\sqrt{2}}(|0_{1}0_{2}\rangle-|1_{1}1_{2}\rangle),~{}|u_{3}\rangle=\frac{1}{\sqrt{2}}(|0_{1}1_{2}\rangle+|1_{1}0_{2}\rangle),~{}|u_{4}\rangle=\frac{1}{\sqrt{2}}(|0_{1}1_{2}\rangle-|1_{1}0_{2}\rangle)\}$. Here the subscript $j=1,2$ on the rhs refers to the qubit $j$, for which

[TABLE]

with $E_{j}$ the overall energy scale of each qubit, $\sigma_{x}$ and $\sigma_{z}$ the Pauli matrices, $2\Delta_{j}$ the dimensionless energy splitting at $q=0$, and $q$ is the flux applied equally to both qubits. We assume that the system is fully symmetric, i.e. $E_{0}\equiv E_{1}=E_{2}$, and $\Delta\equiv\Delta_{1}=\Delta_{2}$ and that all the noise sources and their couplings to the individual qubits are equal. The eigenenergies of the Hamiltonian (in units of $E_{0}$) are given by

[TABLE]

and the corresponding eigenstates are

[TABLE]

For the noise, we consider a generic form of linear coupling between each qubit and the noise source as

[TABLE]

where $\hat{A}_{m}$ determines the coupling and $\delta\hat{X}_{m}(t)$ is the time $t$ dependent fluctuation of the quantity. In what follows we investigate the cases of different degrees of correlation between two noise fluctuators. The noise correlators with the help of their Fourier transform are given by

[TABLE]

where $S(\omega)$ is the unsymmetrized noise spectrum of each current. Here, we define $\chi_{mn}$ as the degree of correlation of noise sources $m$ and $n$. For autocorrelation, $\chi_{11}=\chi_{22}=+1$, and for crosscorrelation we set $\chi_{12}=\chi_{21}=\chi$, where $-1\leq\chi\leq+1$. If $\chi=+1$, noise is fully correlated and for $\chi=-1$ anticorrelated, whereas for $\chi=0$ we have uncorrelated noise from independent sources.

The transition rates from the $k^{th}$ to the $l^{th}$ instantaneous eigenstate due to the noise source(s) can be calculated from the Fermi’s golden rule as

[TABLE]

where $\omega_{kl}=E_{kl}/\hbar=E_{0}(\lambda_{k}-\lambda_{l})/\hbar$. The rates are the off-diagonal elements of $\Gamma_{i\rightarrow j}$ given by

[TABLE]

where $\Gamma_{\Sigma}=\Gamma_{\uparrow}+\Gamma_{\downarrow}$, $\Gamma_{\downarrow,\uparrow}=\Gamma_{\downarrow,\uparrow,{\rm C}}+\Gamma_{\downarrow,\uparrow,{\rm H}}$, and $\Gamma_{\downarrow,\uparrow}=\frac{g^{2}}{\hbar^{2}}\frac{\Delta^{2}}{q^{2}+\Delta^{2}}S(\pm\omega_{0})$ is the corresponding rate for a single qubit with identical parameters coupled to the noise source, $\hbar\omega_{0}=2E_{0}\sqrt{q^{2}+\Delta^{2}}$ is the level spacing, and $g$ is the coupling constant Jukka . The effect of $\chi$ on the transition rates between the energy levels for decoupled qubits is illustrated in Fig. 2. According to (12) for fully (anti)correlated noise $\chi=+1$ ($\chi=-1$), there are forbidden transitions to/from $|2\rangle$ $(|3\rangle)$, and as a result it becomes a protected state as shown in Fig. 2.

To demonstrate the significance of the correlation of noise on measurable quantities, we focus on the system depicted in Fig. 3 for different values of $\chi$. We consider noise sources to be thermal baths. This set-up allows us to investigate quantum heat transport between the baths, a topic of considerable experimental interest currently schwab ; jukkanature ; jezouin ; partanen . In this case, the fluctuating quantity $\delta\hat{X}_{m}(t)$ is presented by electric current noise $\delta i_{m}(t)$ and the coupling $\hat{A}_{m}$ is $g_{m}\sigma_{z,m}$. Here $g_{m}$ is coupling, e.g., by mutual inductance between each qubit and the fluctuating current. As shown in Fig. 3, the two qubits in the middle are coupled to heat baths at two different temperatures $T_{\rm B}$, where ${\rm B}={\rm C},{\rm H}$ indicates the ”cold” and ”hot” baths, respectively. The baths are represented by the resistors $R_{\rm B}$ embedded in the $LC$ resonators with a quality factor $Q_{\rm B}=\sqrt{L_{\rm B}/C_{\rm B}}/R_{\rm B}$.

The quantity of interest here is the power transmitted between the hot and cold baths mediated by the qubits. We study the full system by writing the master equation for the density matrix of the system and the environment $\rho_{\rm tot}=\rho\otimes\rho_{\rm E}$ in the interaction picture of the two qubit system as

[TABLE]

Here we have assumed that the qubits are driven by time-dependent rotation $H_{\rm D,I}=-i\hbar D^{\dagger}\dot{D}$ where $D$ is given by

[TABLE]

$H_{{\rm cS},I}$ arises from the noise described above, and presented specifically for similar set-up in niskanen2007 . The components of $\rho$ can be obtained from the full master equation breuer by tracing out the environment with the result

[TABLE]

Here $\Omega=2\pi\hbar f/E_{0}$ denotes the dimensionless frequency of the drive and $u=2\pi ft$ the time, where $f$ is the actual driving frequency.

The instantaneous power to bath $\rm B$ can then be written as bayan

[TABLE]

In the non-driven case the relaxation towards steady state is governed by

[TABLE]

where $\rho_{d}=(\rho_{11}~{}~{}\rho_{22}~{}~{}\rho_{33}~{}~{}\rho_{44})^{\rm T}$. For $\chi=\pm 1$ the steady-state solution of Eq. (23) depends on the initial condition applied to the system. Due to forbidden transitions according to Eq. (12), the system behaves differently based on its initialization. With $\chi=+1$, if the system is initialized in the state $|2\rangle$ we have $\rho_{22}=1,\rho_{11}=\rho_{33}=\rho_{44}=0$ which demonstrates that $|2\rangle$ is a protected state. On the other hand, initializing in the subspace $\{|1\rangle,|3\rangle,|4\rangle\}$ for $\chi=+1$ leads to $\rho_{22}=0$, $\rho_{11}=\frac{1}{(1+r)^{2}}$, $\rho_{33}=\frac{r}{(1+r)^{2}}$, $\rho_{44}=\frac{r^{2}}{(1+r)^{2}}$, where $r=\Gamma_{\uparrow}/\Gamma_{\downarrow}$. For $\chi=-1$, one should simply swap states $|2\rangle$ and $|3\rangle$ above. Generally for $\chi\neq\pm 1$ we have

[TABLE]

which is independent of the correlation $\chi$. Then the steady state power to bath ${\rm C}$ reads

[TABLE]

where $\rho_{gg}=1-\rho_{ee}$ is the ground state population of a single qubit in the instantaneous eigenbasis and $P_{0}$ the transmitted power by it to the cold reservoir Jukka . The power $P_{\rm C}$ in steady state thus scales with the number of qubits and is independent of the degree of correlation of the noise for $\chi\neq\pm 1$.

The inset of Fig. 4b demonstrates the puzzling result that this power is larger than $2P_{0}$ for the special values $\chi=\pm 1$, i.e. for fully correlated or anticorrelated noise. The origin of this result becomes obvious by looking at the dynamics of the density matrix after the system has been initialized in an arbitrary state. Due to the presence of a protected state $|2\rangle$ ($|3\rangle$) for $\chi=+1$ ($\chi=-1$), the power of Eq. (Correlated vs. uncorrelated noise acting on a quantum refrigerator) cannot be reached in finite time for the case of fully (anti)correlated noise. The time dependence of power $P_{\rm C}$ normalized by the uncorrelated power $2P_{0}$ when the system is initialized in state $|1\rangle$ and $|2\rangle$, respectively, at $t=0$ are shown in Fig. 4a,b. The asymptotic value of $P_{\rm C}$ does not depend on this initial state, except for the cases $\chi=\pm 1$. It is seen here that for $\chi\rightarrow+1$, the time to approach $P_{\rm C}=2P_{0}$ becomes longer and longer, and finally, this relaxation time becomes infinite for $\chi=+1$. This explains the result in the inset of Fig. 4b, where power is enhanced for $\chi=\pm 1$ above that of the other values of $\chi$. The same slow relaxation for higher values of $\chi$ is seen in the inset of Fig. 4a, where we plot the populations of the states $\rho_{ii}$ $(i=1,2,3,4)$ when the system is initialized in $|2\rangle$ for two different values of $\chi=0$ and $0.9$. It is also worth mentioning that if one initializes the system to the state $|2\rangle$ for $\chi=+1$ or $|3\rangle$ for $\chi=-1$, the power $P_{\rm C}=0$, as this state is a protected one. For any other value of $\chi$ away from $\chi=\pm 1$ the power approaches $2P_{0}$ after a sufficiently long time also in this case.

We discuss finally a quantum Otto refrigerator Jukka ; niskanen2007 . In this device, applying periodic time dependent drive to the qubits in Fig. 3, heat can be transferred from the cold bath to the hot one, provided $\omega_{\rm H}=1/\sqrt{L_{\rm H}C_{\rm H}}>\omega_{\rm C}=1/\sqrt{L_{\rm C}C_{\rm C}}$. We introduce a standard driving protocol $q(t)=(1+\cos 2\pi ft)/4$. In the numerical results, the power is averaged over one cycle once it has reached the steady state. Solving the general master equation (21) numerically, we plot the frequency dependence of the cooling power of the quantum refrigerator for different degrees of correlation in two different frequency ranges in Fig. 5a,b. It is vivid that at very low frequencies the curves for all values of $\chi$ (except $\chi=+1$) in Fig. 5a collapse on each other. They start to deviate from the curve at $\chi=0$ at the critical frequency $\Omega_{c}\propto(1-\chi)$. This is because of the competition between the slowest transition rates $\propto(1-\chi)$ in Eq. (12) to/from $|2\rangle$ and the driving frequency $\Omega$. At higher frequencies, $\Omega\gg\Omega_{c}$, the transitions to/from state $|2\rangle$ cannot follow the drive (Fig. 5b), $|2\rangle$ is thus dynamically protected and we effectively deal with a three-level system. Thus for $\Omega\gg\Omega_{c}$ , all the curves with $\chi\sim+1$ converge to the same value. In this regime $\rho_{22}$ has a small but essentially time-independent value. For $\chi=+1$, $\rho_{22}=0$ as the system was initialized in state $|1\rangle$. For this particular value of $\chi$ the power $-P_{\rm C}$ is again higher than in the partially correlated case.

In conclusion, we have investigated the golden-rule transition rates between the four energy levels of a two-qubit system when it is subjected to fully and partially (anti)correlated noise sources. By tuning the degree of correlation of the noise sources, we demonstrate protected states and variations in the transmitted power between thermal baths. This power exhibits a different steady-state value in the presence of a protected state as opposed to that of the standard four level system. In particular, the former power vanishes when the qubits are initialized in a protected state. Moreover, for nearly (anti)correlated noise, there is a long relaxation time to reach the steady state level of power which is fully independent of the level of correlation of the noise for $\chi\neq\pm 1$. Under AC driven conditions, there is an interesting interplay between this slow relaxation rate and the driving frequency, which governs the power of a quantum refrigerator that we present as an example.

We acknowledge Yuri Galperin, Rosario Fazio and Michele Campisi for useful discussions. The work was supported by the Academy of Finland (grants 272218 and 284594).