Assessment of weak-coupling approximations on a driven two-level system under dissipation

The standard weak-coupling approximations associated to open quantum systems have been extensively used in the description of a two-level quantum system, qubit, subjected to relatively weak dissipation compared with the qubit frequency. However, recent progress in the experimental implementations of controlled quantum systems with increased levels of on-demand engineered dissipation has motivated precision studies in parameter regimes that question the validity of the approximations, especially in the presence of time-dependent drive fields. In this paper, we address the precision of weak-coupling approximations by studying a driven qubit through the numerically exact and non-perturbative method known as the stochastic Liouville-von Neumann equation with dissipation. By considering weak drive fields and a cold Ohmic environment with a high cutoff frequency, we use the Markovian Lindblad master equation as a point of comparison for the SLED method and study the influence of the bath-induced energy shift on the qubit dynamics. We also propose a metric that may be used in experiments to map the regime of validity of the Lindblad equation in predicting the steady state of the driven qubit. In addition, we study signatures of the well-known Mollow triplet and observe its meltdown owing to dissipation in an experimentally feasible parameter regime of circuit electrodynamics. Besides shedding light on the practical limitations of the Lindblad equation, we expect our results to inspire future experimental research on engineered open quantum systems, the accurate modeling of which may benefit from non-perturbative methods.

Time delay in the strong field limit for null and timelike signals and its simple interpretation

Gravitational lensing can happen not only for null signals but also timelike signals such as neutrinos and massive gravitational waves in some theories beyond GR. In this work we study the time delay between different relativistic images formed by signals with arbitrary asymptotic velocity v in general static and spherically symmetric spacetimes. A perturbative method is used to calculate the total travel time in the strong field limit, which is found to be a quasi-power series of the small parameter $$a=1-b_c/b$$ a = 1 - b c / b where b is the impact parameter and $$b_c$$ b c is its critical value. The coefficients of the series are completely fixed by the behaviour of the metric functions near the particle sphere $$r_c$$ r c and only the first term of the series contains a weak logarithmic divergence. The time delay $$\Delta t_{n,m}$$ Δ t n , m to the leading non-trivial order was shown to equal the particle sphere circumference divided by the local signal velocity and multiplied by the winding number and the redshift factor. By assuming the Sgr A* supermassive black hole is a Hayward one, we were able to validate the quasi-series form of the total time, and reveal the effects of the spacetime parameter l, the signal velocity v and the source/detector coordinate difference $$\Delta \phi _{sd}$$ Δ ϕ sd on the time delay. It is found that as l increases from 0 to its critical value $$l_c$$ l c , both $$r_c$$ r c and $$\Delta t_{n,m}$$ Δ t n , m decrease. The variation of $$\Delta t_{n+1,n}$$ Δ t n + 1 , n for l from 0 to $$l_c$$ l c can be as large as $$7.2\times 10^1$$ 7.2 × 10 1 [s], whose measurement then can be used to constrain the value of l. While for ultra-relativistic neutrino or gravitational wave, the variation of $$\Delta t_{n,m}$$ Δ t n , m is too small to be resolved. The dependence of $$\Delta t_{n,-n}$$ Δ t n , - n on $$\Delta \phi _{sd}$$ Δ ϕ sd shows that to temporally resolve the two sequences of images from opposite sides of the lens, $$|\Delta \phi _{sd}-\pi |$$ | Δ ϕ sd - π | has to be larger than a certain value, or equivalently if $$|\Delta \phi _{sd}-\pi |$$ | Δ ϕ sd - π | is small, the time resolution of the observatories has to be good.

Analytical and Experimental Analyses of Nonlinear Vibrations in a Rotary Inverted Pendulum

Gaining insight into possible vibratory responses of dynamical systems around their stable equilibria is an essential step, which must be taken before their design and application. The results of such a study can significantly help prevent instability in closed-loop stabilized systems through avoiding the excitation of the system in the neighborhood of its resonance. This paper investigates nonlinear oscillations of a Rotary Inverted Pendulum (RIP) with a full-state feedback controller. Lagrange’s equations are employed to derive an accurate 2-DoF mathematical model, whose parameter values are extracted by both the measurement and 3D modeling of the real system components. Although the governing equations of a 2-DoF nonlinear system are difficult to solve, performing an analytical solution is of great importance, mostly because, compared to the numerical solution, the analytical solution can function as an accurate pattern. Additionally, the analytical solution is generally more appealing to engineers because their computational costs are less than those of the numerical solution. In this study, the perturbative method of multiple scales is used to obtain an analytical solution to the coupled nonlinear motion equations of the closed-loop system. Moreover, the parameters of the controller are determined, using the results of this solution. The findings reveal the existence of hardening- and softening-type resonances at the first and second vibrational modes, respectively. This led to a wide frequency range with moderately large-amplitude vibrations, which must be avoided when adjusting a time-varying set-point for the system. The analytical results of the nonlinear vibration of the RIP are verified by experimental measurements, and a very good agreement is observed between the results of both approaches.

Finding gene network topologies for given biological function with recurrent neural network

Searching for possible biochemical networks that perform a certain function is a challenge in systems biology. For simple functions and small networks, this can be achieved through an exhaustive search of the network topology space. However, it is difficult to scale this approach up to larger networks and more complex functions. Here we tackle this problem by training a recurrent neural network (RNN) to perform the desired function. By developing a systematic perturbative method to interrogate the successfully trained RNNs, we are able to distill the underlying regulatory network among the biological elements (genes, proteins, etc.). Furthermore, we show several cases where the regulation networks found by RNN can achieve the desired biological function when its edges are expressed by more realistic response functions, such as the Hill-function. This method can be used to link topology and function by helping uncover the regulation logic and network topology for complex tasks.

Perturbative deflection angle, gravitational lensing in the strong field limit and the black hole shadow

A perturbative method to compute the deflection angle of both timelike and null rays in arbitrary static and spherically symmetric spacetimes in the strong field limit is proposed. The result takes a quasi-series form of $$(1-b_c/b)$$ ( 1 - b c / b ) where b is the impact parameter and $$b_c$$ b c is its critical value, with coefficients of the series explicitly given. This result also naturally takes into account the finite distance effect of both the source and detector, and allows to solve the apparent angles of the relativistic images in a more precise way. From this, the BH angular shadow size is expressed as a simple formula containing metric functions and particle/photon sphere radius. The magnification of the relativistic images were shown to diverge at different values of the source-detector angular coordinate difference, depending on the relation between the source and detector distance from the lens. To verify all these results, we then applied them to the Hayward BH spacetime, concentrating on the effects of its charge parameter l and the asymptotic velocity v of the signal. The BH shadow size were found to decrease slightly as l increases to its critical value, and increase as v decreases from light speed. For the deflection angle and the magnification of the images however, both the increase of l and decrease of v will increase their values.

Dual path integral: a non-perturbative approach to strong coupling

We develop a non-perturbative method for calculating partition functions of strongly coupled quantum mechanical systems with interactions between subsystems described by a path integral of a dual system. The dual path integral is derived starting from non-interacting subsystems at zeroth order and then by introducing couplings of increasing complexity at each order of an iterative procedure. These orders of interactions play the role of a dual time and the full quantum partition function is expressed as a transition amplitude in the dual system. More precisely, it is expressed as a path integral from a deformation-operators dependent initial state at zero time/order to the inverse-temperature dependent final state at later time/order. We provide examples of strongly coupled systems with up to first-order interactions (e.g. Ising model) and arbitrary high-order interactions (e.g. $$1+1\hbox {D}$$ 1 + 1 D QFT). We also discuss a possible emergence of space-time, quantum field theories and general relativity in context of the dual path integral.

Regge amplitudes in generalized fishnet and chiral fishnet theories

We extend the analysis of [1] to study the Regge trajectories of the Mellin amplitudes of the 0- and 1-magnon correlators of the generalized Fishnet theory in d dimensions and one type of correlators of chiral fishnet theory in 4 dimensions. We develop a systematic procedure to perturbatively study the Regge trajectories and subsequently perform the spectral integral. Our perturbative method is very generic and in principle can be applied to correlators whose perturbative Regge trajectories obey some structural conditions which we list down. Our d dimensional results reduce to previously known results in d = 4 for 0-magnon and 1-magnon. As a non-trivial check, we show that the results for 1-magnon correlator in d = 8, when evaluated using the exact techniques in [1, 2] are in perfect agreement with our d dimensional perturbative results. We also perturbatively compute the Regge trajectories and Regge-Mellin amplitudes of the chiral fishnet correlator $$ \left\langle \mathrm{Tr}\left[{\phi}_1\left({x}_1\right){\phi}_1\left({x}_2\right)\right]\mathrm{Tr}\left[{\phi}_1^{\dagger}\left({x}_3\right){\phi}_1^{\dagger}\left({x}_4\right)\right]\right\rangle $$ Tr ϕ 1 x 1 ϕ 1 x 2 Tr ϕ 1 † x 3 ϕ 1 † x 4 using the techniques developed in this paper. Since this correlator has two couplings κ and ω, we have obtained closed-form results in the limit κ → 0, ω → 0 with κ/ω held constant. We verify this computation with an independent method of computing the same and obtain perfect agreement.

Constraining the spacetime spin using time delay in stationary axisymmetric spacetimes

Total travel time t and time delay $$\Delta t$$ Δ t between images of gravitational lensing (GL) in the equatorial plane of stationary axisymmetric (SAS) spacetimes for null and timelike signals with arbitrary velocity are studied. Using a perturbative method in the weak field limit, t in general SAS spacetimes is expressed as a quasi-series of the impact parameter b with coefficients involving the source-lens distance $$r_s$$ r s and lens-detector distances$$r_d$$ r d , signal velocity v, and asymptotic expansion coefficients of the metric functions. The time delay $$\Delta t$$ Δ t to the leading order(s) were shown to be determined by the spacetime mass M, spin angular momentum a and post-Newtonian parameter $$\gamma $$ γ , and kinematic variables $$r_s,~r_d,~v$$ r s , r d , v and source angular position $$\beta $$ β . When $$\beta \ll \sqrt{aM}/r_{s,d}$$ β ≪ aM / r s , d , $$\Delta t$$ Δ t is dominated by the contribution linear to spin a. Modeling the Sgr A* supermassive black hole as a Kerr–Newman black hole, we show that as long as $$\beta \lesssim 1.5\times 10^{-5}$$ β ≲ 1.5 × 10 - 5 [$$^{\prime \prime }$$ ″ ], then $$\Delta t$$ Δ t will be able to reach the $$\mathcal {O}(1)$$ O ( 1 ) second level, which is well within the time resolution of current GRB, gravitational wave and neutrino observatories. Therefore measuring $$\Delta t$$ Δ t in GL of these signals will allow us to constrain the spin of the Sgr A*.