Spectral origin of non-Markovian open-system dynamics: A finite harmonic model without approximations

Dissipation caused by nonhomogeneous bosonic finite chain environments (implementable in, e.g., segmented ion traps) is investigated through an exact diagonalization approach, avoiding common approximations used to describe open systems. Different spectral densities, including band gaps, can be engineered to separately assess different factors leading to memory effects, namely, the environment's size, temperature, proximity of the cutoff frequency, the spectral density's shape (sub-Ohmic, Ohmic, super-Ohmic), and the strength of its coupling to the system. Non-Markovianity is quantified with two recently introduced measures related to information backflow and nondivisibility of the system dynamical map. By sweeping the bath spectrum via tuning of the system frequency we show strongest memory effects at band-gap edges and provide an interpretation based on energy flow between system and environment. A system weakly coupled to a stiff chain ensures a Markovian dynamics, while the size of the environment as well as the local density of modes are not substantial factors. We show an opposite effect when increasing the temperature inside or outside the spectral band gap. Further, non-Markovianity arises for larger (negative and positive) powers of algebraic spectral densities, being the Ohmic case not always the most Markovian one.

Figure 1

Examples of spectral densities which can be generated in our model: Rubin chain with $g=0.1$ and $k=0.01$ (black dashed line), gapped spectral density with $g=0.1$, $h=0.05$, and $k=0.0075$ (red continuous line), and increasing coupling density with $g_i=0.1+0.05i$ and $k=0.01$ (blue dotted line); all cases with $N=200$ oscillators. Bath local potential frequency ${\omega }_0=0.2$. Couplings and frequencies are specified relative to a fixed (arbitrary) frequency unit, with $\hslash =m=1$, in this and following figures. Schematic for the three plotted examples (top left) and of a star configuration (top right) also represented.

Figure 2

(a) Averaged fidelity as a function of system-bath coupling $k$ and bandwidth $\delta$, for a band-gap density [shaded area in (b)] with $g=0.1$, $h=0.05$, ${\omega }_0=0.3$, and ${\omega }_S=0.65$ at temperature $T=0$. Integration performed up to ${\tau }_F=400$ for $N=50$ oscillators. (b) System excitation number $n=\langle H_S\left({\tau }_F\right)\rangle /{\omega }_S$ at ${\tau }_F=400$ as a function of the system frequency ${\omega }_S$ for a band-gap model with $g=0.1$, $h=0.05$, and ${\omega }_0=0.3$ [$J\left(\omega \right)$ as shaded area] at $T=0$. System-bath coupling $k=0.005$ (red solid) and $k=0.025$ (blue dashed). (Arbitrary units as specified in the caption of Fig. 1.)

Figure 3

BLP non-Markovianity [blue (dot-dashed)] and RHP [black (dashed)] for $N=40$ oscillators, $g=0.2$, $h=0.05$, $k=0.001$, ${\omega }_0=0.5$ when ${\omega }_S$ is swept, for $T=0$. The red (continuous) curve represents the spectral density function used in arbitrary units. The extra dots are evaluated at ${\omega }_S=0.45$ increasing the two-mode squeezing parameter from $r=1$ to $1.5$ (for RHP NM, upper dot) and increasing the squeezing parameter range for the BLP NM (lower dot) maximization (from $r\in \{0.5,1\}$ to $r\in \{0.125,1.125\}$) (see Appendix pp3). (Arbitrary units as specified in the caption of Fig. 1.)

Figure 4

Left: log$\left(M_{\text{BLP}}\right)$ for 50 oscillators, $g=0.1$, $h=0.05$, $k=0.001$, and ${\omega }_0=0.3$, for temperatures $T\in [0,10]$ and ${\omega }_S$ (the limits of the acoustic and optical bands are evident from the discontinuity of $M_{\text{BLP}}$). Middle: Temperature dependence of $M_{\text{BLP}}$ (black) and $M_{\text{RHP}}$ [red (gray)] for ${\omega }_S=0.25\left(0.375\right)$ with continuous (dashed) line, i.e., out of (in) resonance. Right: $M_{\text{BLP}}$ (continuous) and $M_{\text{RHP}}$ (dashed) at ${\omega }_S=0.575$ for a homogeneous 50 oscillator (Rubin) chain, $g=h=0.1$ (i.e., no gap) and ${\omega }_0=0.3$. (Arbitrary units as specified in the caption of Fig. 1.)

Figure 5

$M_{\text{BLP}}$ for the star configuration with equally spaced eigenfrequencies, and couplings designed to give a specific $J\left({\omega }_S\right)$. Left: spectral density function $J\left(\omega \right)=k{(\omega /{\omega }_S)}^l\Theta (\omega -0.25)\Theta (0.75-\omega )$ ($\Theta$ is the step function) is used. We show $M_{\text{BLP}}$ as a function of $l$ at ${\omega }_S=0.5$, for temperatures $T=0,0.25,0.5,1$. Right: spectral density function $J\left(\omega \right)=k\omega \sqrt{{\omega }_c^2-{\omega }^2}/{\omega }_c$ is used with 40 oscillators and $k=0.00001$. We plot $M_{\text{BLP}}$ at ${\omega }_S=0.4$ when varying ${\omega }_c$. (Inset shows some of the $J\left(\omega \right)$ functions corresponding to this situation; arbitrary units as specified in the caption of Fig. 1.)

Figure 6

RHP non-Markovianity against system frequency ${\omega }_S$ comparing the result with $r=1,2,3$ (continuous, dashed, dashed-dashed). The behavior is the same, only with higher absolute values. (Arbitrary units as specified in the caption of Fig. 1.)

Figure 7

BLP non-Markovianity as used for the figures related to temperature, for $T=0,0.5,1$ and ${\omega }_S=0.375$ ($r_1=1$). It can be seen that for higher temperatures several maxima appear at different squeezing parameters $r_2,{\theta }_2$. (Arbitrary units as specified in the caption of Fig. 1.)

Figure 8

Top: BLP non-Markovianity against system frequency ${\omega }_S$ with $J\left(\omega \right)=k{(\omega -{\omega }_0)}^l\Theta ({\omega }_c-\omega )$, ${\omega }_0=1$, ${\omega }_c=5$, $k=0.00001$, and $l=1,2,3,4$ (black for $l=1$ to gray for $l=4$. Bottom: BLP NM versus $l$ evaluated at ${\omega }_S=1,3,5$ (sparse dashed, tight dashed, and continuous, respectively) for the spectral density above. (Arbitrary units as specified in the caption of Fig. 1.)

Figure 9

BLP non-Markovianity against system frequency ${\omega }_S$ with $J\left(\omega \right)=k{[(\omega -{\omega }_0)/({\omega }_c-{\omega }_0)]}^l\Theta ({\omega }_c-\omega )$, ${\omega }_0=3$, ${\omega }_c=5$, $k=0.00001$, and $l=1/2,1,2$ (continuous, dashed, and dotted, respectively). We have drawn the spectral density to guide the eye. Inset: BLP NM versus $l$ evaluated at ${\omega }_S=4$. (Arbitrary units as specified in the caption of Fig. 1.)