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How Useful are Computational Models to Traders in Their Decision Making?

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Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: References

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References Appendix D: QMPE in region (e): 3rd order EP The region (e) in the (d,˜ Γ) plane of Hatano’s model is composed of a single point It is a third order EP where all the non-zero eigenvalues become equal and their eigenvectors coalesce. Since this is a single point with fixed values of ˜d and Γ, the right eigenvectors are explicitly given by Of course |r1i and hℓ1| have same expressions both in Eqs. (3) and (D2). The left eigenvectors are ) The non-zero eigenvalue of the Lindbladian [Eq. (2)] in region (e) is −iλ = −i4√ 3. Importantly, as a consequence of the 3rd order EP, we can only achieve an approximately diagonal form i.e. Jordan normal form LbJn leading to the equation Due to the off-diagonal nonzero defects in Eq. (D5), the time evolution of density matrix elements and other observables pick up additional algebraic terms in time. 1. Density matrix elements In region (e), the density matrix elements have the expressions below In Fig. 15(a), we present existence of t˜+ and t˜− as function of the control parameters ˜dI and ˜dII using the fixed dissipation protocol. The solutions’ surfaces in Fig. 15(a) reveal that there are regions where only one of the solutions t˜+ or t˜− exist leading to single QMPE, and in other regions both the solutions exist resulting in double QMPE. Particularly, lower values of ˜dI and ˜dII correspond to single intersection time solution and therefore single QMPE. To understand the behaviors of t˜+,− more clearly, we fix ˜dI and display t˜+,− as functions of ˜dII in Fig. 15(b). The Fig. 15(b) shows that the two branches t˜+ and t˜− interchange as ˜dI < ˜dII. More specifically, for ˜dI > ˜dII we have t˜+ > t˜− and for ˜dI < ˜dII we have t˜+ < t˜−. The underlying reason is that the discriminant p γ 2 1 − 4γ2γ0 in Eq. (D8) is proportional to sign of ( ˜dI − ˜dII). Thus the two intersection times interchange their roles by changing the sign accompanying the discriminant as we go from ˜dI > ˜dII to ˜dI < ˜dII. The Fig. 16(a) demonstrates an explicit example of double QMPE in ground state probability in region (e). Like other regions, here also the second intersection (shown in the inset of Fig. 16(a)) is much feeble in comparison to the first intersection. 2. Average energy For the same parameters used to obtain QMPE for ρgg(t) in Fig. 16(a), we would like to examine the possibility of QMPE in average energy E(t). Indeed, we observe in Fig. 16(b) that energy also exhibits double QMPE, marked by the twice intersection of zero by the energy difference ∆E(t) between the two copies. The insets of Fig. 16(b) shows separately the two intersections in the time evolution of the energies for the two copies. The second intersection is much weaker in magnitude compared to the first intersection just like the case of ground state probability. Note that the first intersection between the energies of I and II occur after they overshoot their steady state value which they reach during some early time of the relaxation process. 3. von Neumann entropy Here we investigate the von Neumann entropy SvN(t), starting from two copies I and II. We use same parameters that have produced double QMPE both in ground state probability [Fig. 16(a)] and average energy [Fig. 16(b)]. In fact, the Fig. 17(a) shows that the entropy also exhibits double QMPE indicated by twice zero intersections (the second one shown in inset of Fig. 17(a)) of the entropy difference ∆SvN(t) between the two copies. 4. Temperature In this section we study the thermal QMPE in region (e). We observe thermal QMPE in region (e) by using variable dissipation protocol in Fig. 17(b). There we observe single intersection and reversal of identities (i.e. hotter becoming colder and vice versa) occur after the negative temperatures have happened. So, such temperatures can be considered as well-defined and therefore this intersection in Fig. 17(b) corresponds to single QMPE in temperature in region (e).

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Appendix D: QMPE in Region (e)

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References Appendix C: QMPE in region (b): purely exponential relaxation In this Appendix we discuss QMPE in region (b) of Hatano’s model that contain eigenvalues 0, −iλ2, −iλ3, −iλ4 obeying 0 < λ2 < λ3. 1. Density matrix elements where In Fig. 12(a), we present the intersection time solutions denoting QMPE in ρgg(t) as functions of the control parameters ˜dI and ˜dII. Indeed, we observe that there are parameter regions where both t˜+ and t˜− are positive and finite, indicating double QMPE. Other regions have only one intersection time, either t˜+ and t˜−, implying single QMPE. To demonstrate double QMPE, the Fig. 12(b) exhibits a specific example of double intersection in ρgg(t). However, the two copies become close enough so that the second intersection (shown in inset of Fig. 12(b)) is weak in magnitude and hard to detect. 2. Energy We look for solutions to ∆Es(t) = 0, which can give to single or multiple QMPE depending on the number of positive finite intersection time solutions. The Fig. 13(a) shows that the average energy exhibits double QMPE indicated by the twice zero intersections of ∆Es(t), for the same set of parameter values producing double QMPE in ρgg(t) in Fig. 12(b). 3. von-Neumann entropy Here we ask if the entropy, like density matrix and energy, also exhibit double QMPE for the same set of parameters used in Figs. 12(b) and 13(a). However, in Fig. 13(b), we observe that the entropy difference ∆SvN(t) between two copies intersect zero only once, giving rise to single QMPE instead of double QMPE. 4. Temperature In Fig. 14(a), we present the time evolution of temperatures for the two copies I and II, one initially starting as hotter with respect to the other. We observe that the temperatures of the two copies intersect each other i.e. the hotter becomes colder and vice versa. Notably, unlike ground state probability [Fig. 12(b)], energy [Fig. 13(a)] that can exhibit double QMPE, we have observed only single QMPE in temperature. If observation of multiple QMPE is possible or not with purely exponential relaxations, remains an open question. 5. Kullback-Leibler divergence Here we study the possibility of QMPE in KL divergence for the same parameters used to explore thermal QMPE in Figs. 14(a). The Fig. 14(b) shows QMPE in KL divergence, however the intersection time corresponds to negative temperatures in Fig. 14(a).

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Appendix C: QMPE in Region (b)

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References Appendix B: Eigenvectors of L in region (a1) (a1) In this Appendix, we give the explicit expressions for the right eigenvectors |rki and left eigenvectors hℓk| of the Lindbladian [Eq. (2)] in region (a1) [Sec. V], where k = 2, 3, 4. The right and left eigenvectors corresponding to the zero eigenvalue i.e. steady state. have already been provided in Eq. (3). Below are the expressions for the right eigenvectors The left eigenvectors are given by where

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Appendix B: Eigenvectors

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References VI. SUMMARY We have studied QMPE in a two-level quantum system subjected to an oscillatory electric field and dissipative coupling with the environment. The Lindbladian of this two-level driven dissipative system, termed as Hatano’s model, has rich variety of eigen-spectrum, especially EPs and complex eigenvalues which are generic features of non-Hermitian quantum systems. We have investigated the roles of complex eigenvalues and EPs on the QMPE. The exact analytical formulae obtained for time evolved density matrix elements enable us to propose analytical criteria for the QMPE in observables. Interestingly, at the EPs, the density matrix elements and consequently observables (e.g. energy) attain additional algebraic time dependent relaxations apart from the usual exponential relaxations. Such additional algebraic time dependence at the EPs leads to double QMPE (twice intersections between trajectories starting at different initial conditions and relaxing towards the same steady state). We have procured the exact expressions for intersection times of trajectories, that mark the occurrence of QMPE and we provide analytical criteria that demarcate control parameters’ regimes with single or double or no QMPE. Whereas the multiple QMPE at EPs in the present model is restricted to maximum double QMPE i.e. two intersections, the multiple QMPE at the complex eigenvalues can result in even more number of intersections in a oscillatory manner. Notably, we have established the existence of thermal QMPE (single QMPE at EPs and multiple QMPE at complex eigenvalues), in presence of coherence in Hatano’s model. It is fascinating that, similar to energy and entropy, temperature also exhibits multiple QMPE where the initially hotter and colder copies intersect each other multiple times. Such multiple reversals of identities (i.e. hotter becoming colder and vice versa) result in multiple thermal QMPE. In spite of achieving multiple QMPE in several observables (ground state probability, energy, entropy, temperature) using either additional algebraic time dependence at EPs or oscillations at complex eigenvalues, surprisingly the KL divergence (a distance measure from steady state) shows single QMPE only. Overall, our theoretical analysis exhibits that EPs and complex eigenvalues can be used as systematic routes to generate multiple QMPE. However, we should mention that the presence of EPs or oscillations are not necessary for multiple QMPE. In this connection, we have shown that multiple QMPE can be achieved even with purely exponential relaxations, by putting suitable constraints on the gap statistics of the eigen-spectrum (Appendix C). An important and generic future goal in QMPE research would be to understand its utility in speeding up quantum processes. In fact, a recent work on accelerating relaxation in open quantum systems by engineering EPs [84] hints towards the connection of such acceleration to QMPE. Our present work analyzes in detail how EPs lead to QMPE and provide criteria on control parameters to observe the QMPE. It is important to find out the deeper connections between the QMPE and faster relaxations at EPs and for generic open quantum systems. In this connection, an intriguing question is to find how the quantum speed limit may affect the degree of faster relaxation that can be achieved through the QMPE. Our work on QMPE by controlling the amplitudes of oscillatory electric fields hints towards possible experimental realizations of the QMPE in qubit, keeping in mind the recent advancement in qubit related experiment [119]. The analysis of QMPE in the present study paves path for future investigations of QMPE in quantum many-body systems. - We thank Naomichi Hatano, Sosuke Ito, Ryo Hanai, Arnab Pal, Naruo Ohga and Kohei Yoshimura for useful discussions. This work is partially supported by the Grants-in-Aid for Scientific Research (Grant No. 21H01006 and No. 20K14428). A.K.C. gratefully acknowledges postdoctoral fellowship from the YITP. The numerical calculations have been done on Yukawa-21 at the YITP. Acknowledgements.

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Summary

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References V. QMPE IN REGION (a1): OSCILLATIONS In this section, we explore QMPE in region (a1) of Hatano’s model which contains complex eigenvalues with both non-zero real and imaginary parts. The eigenvalues are non-degenerate and they are denoted by 0, λ2, λ3 = λre+iλim, λ4 = λre−iλim, where the subscripts re and im correspond to real and imaginary parts, respectively. In addition, the eigenvalues satisfy the relation 0 < λ2 < λre meaning λ2 is the slowest relaxation mode. In contrast to the nontrivial approximate diagonalization [Eqs. (10)- (11)] for second-order EP in Sec. IV, here the complete diagonalization can be obtained through the usual method A. Density matrix elements The the individual density matrix elements ρj (t) (j = 1, 2, 3, 4) are given by To discuss QMPE in the density matrix, let us focus on the ground state probability i.e. ρgg(t). Of course, there is no algebraic time dependence in the relaxation observed in Sec. IV. Rather, the interesting part of the relaxation in region (a1) originates from the complex nature of the eigenvalues and the corresponding coefficients ak in Eq. (27). Specifically, we have found that a3 and a4 are complex conjugates of each other, they take the forms a3 = are + iaim and a4 = are − iaim. Using such expressions, we obtain the following formula for ρgg(t): In Fig. 7 we present an example of multiple QMPE in the ground state density matrix element ρgg(t), where we have chosen parameters that satisfy the QMPE criteria Eq. (30) for region (a1). We observe that the two copies I and II relax in oscillatory manner and intersect each other multiple times. The inset explicitly shows that the difference between the two copies ∆ρgg(t) intersects zero multiple times, confirming the multiple QMPE. Comparing Fig. 3(a) [region (d)] with the inset of Fig. 7 [region (a1)], we understand that the number of intersections in region (a1) is not restricted to two as in region (d). Therefore, in region (a1) one can have many more intersections and higher order QMPE compared to the maximum double QMPE in the region (d). Since the amplitudes of the oscillations can be controlled by tuning the initial driving strengths ˜dI and ˜dII, complex eigenvalues seem to be useful to obtain multiple intersections and thereby multiple QMPE. It would be important in the future to find the exact number of solutions to ∆ρgg(t) = 0 either analytically or numerically. B. Energy The formal expression for average energy E(t) is given by Eq. (18). In the previous Sec. IV, we have already shown that QMPE in ground state probability does not necessarily imply QMPE in energy, in general. However, it is possible to find control parameters that can give rise to QMPE in both observables, although the intersection time(s) are generally different. Following the same path, we would like to see if E(t) shows multiple QMPE for the same set of parameters used to achieve multiple QMPE for ρgg(t) in Fig. 7. Indeed, Fig. 8 exhibits multiple intersections between time evolved copies of energies starting from I and II, both relaxing towards the same final energy. The inset of Fig. 8 alternatively demonstrates the multiple QMPE through multiple intersections of the energy difference ∆E(t) = EI(t)−EII(t) with zero, remarkably different from Fig. 3(b) with a maximum of two such intersections in region (d). C. von Neumann entropy Next we consider the von Neumann entropy SvN(t) [Eq. (22)]. To observe QMPE in SvN(t), we consider the intersection times for the entropy difference between two copies ∆SvN(t) = S I vN − S II vN. In Fig. 9, we present the behavior of ∆SvN(t) for the same set of parameter values used in Figs. 7 and 8. Indeed, we observe that ∆SvN(t) intersects zero multiple times showing the existence of multiple QMPE in the von Neumann entropy. D. Temperature Here, we study the thermal QMPE in region (a1) using the system temperature defined in Eq. (23). In Fig. 10, we present the time evolution of the temperatures of two copies I and II, one initially hotter than the other. It is fascinating that the temperatures of both the copies relax in oscillatory manner towards the same steady state temperature and during the relaxations, they intersect each other multiple times, leading to multiple thermal QMPE. This, in turn, implies that the hotter and colder copies reverse their identities multiple times (after each intersection the previous hotter copy turns into colder copy and remains so until the next intersection). Such oscillatory multiple thermal QMPE is observed here for the first time in quantum systems (as far as our knowledge is concerned), although multiple thermal MPE in classical inertial suspensions has already been reported [42]. The physical interpretation and utility of multiple thermal QMPE have to be explored in future in more details. Note that we have used the same parameter set for multiple thermal QMPE as in Figs. 7-9. Similar to Ref. [65], here also we observe negative temperature [114– 118] in the early stage of relaxation. This is due to nonmonotonic natures of energy and entropy as functions of time. However, it is satisfying that even if we neglect such initial negative temperatures, we observe multiple QMPE at later times of the relaxation process where the temperatures of both the copies remain positive. E. Kullback-Leibler divergence In this subsection we explore the QMPE in KL divergence defined in Eq. (24). Since we have been able to obtain multiple QMPE in region (a1) in all other observables considered here, it is natural to ask about the possibility of multiple QMPE in DKL(t) as well. However, we have found that the KL divergence does not show any intersection between two copies I and II for the same parameter values used in Figs. 7-10. Therefore, we have to use different values of the initial drives (still within the fixed dissipation protocol) to achieve QMPE in DKL(t), presented in Fig. 11(a). Intriguingly, we have only observed single intersection between the two time evolved copies, resulting in single QMPE in KL divergence. This means that once the initially higher distant copy intersects the initially lower distant copy producing QMPE, thereafter the new lower distant copy (that was initially higher distant) remains closer to the steady state for the rest of the relaxation process. In fact, for all other parameter sets that we have checked, we could not find multiple QMPE in DKL(t). The proof of the possibility or impossibility of multiple QMPE in KL divergence in presence of complex eigenvalues remains unsolved and could be an important question to investigate later. In this scenario, it would be interesting to examine the corresponding behaviors of the speeds vKL(t) [Eq. (25)] of the two copies I and II. In Fig. 11(b), we present the time evolution of speeds for the same set of parameters that gave rise to single QMPE in DKL(t). It is fascinating to observe that the speeds, unlike the distances, intersect each other multiple times producing multiple QMPE. The first intersection between the speeds of the two copies [Fig. 11(b)] shows a time lag in comparison to the intersection between distances of the same copies [Fig. 11(a)]. Such a time lag between the QMPE in KL divergence and the speed is in agreement with the observation in region (d) [Fig. 6]. However, contrary to single QMPE in vKL(t) in region (d), we have obtained multiple QMPE in the speed in region (a1). The contrasting behavior of single QMPE in DKL(t) and multiple QMPE in vKL(t) imply that the initially higher distant copy that becomes lower distant after the intersection in distances, can remain at lower distance even if its speed sometimes become smaller than the speed of the current higher distant copy. This is interesting because the system almost behaves like an intelligent and efficient runner in a race who knows how to adjust the speed (i.e. sometimes running faster and sometimes running slower than the people behind) and still remains in the first position to finish the race.

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: QMPE in Region (a1)

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References Appendix A: Eigenvectors of L in region (d) In this Appendix, we give the explicit expressions for the right eigenvectors |rki and left eigenvectors hℓk| of the Lindbladian [Eq. (2)] in region (d) [Sec. IV], where k = 2, 3, 4. The right and left eigenvectors corresponding to the zero eigenvalue i.e. the steady state. have already been provided in Eq. (3). Below are the expressions for the right eigenvectors where the superscript T denotes transpose. The left eigenvectors are obtained to be where the elements lkn are explicitly given by where the eigenvalues λ2 and λ4 are given by Eq. (12) and the parameters η, δ1, δ2, δ3 used in Eq. (A3) have the following forms

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Appendix A: Eigenvectors

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References III. DIFFERENT REGIMES IN d˜− Γ˜ PLANE AND MULTIPLE QMPE Indeed, in the space of control parameters ( ˜dI, Γ˜ I, ˜dII, Γ˜ II; ˜d, Γ), we find ˜ multiple QMPE where the two trajectories corresponding to some observable, intersect each other multiple times before reaching the same steady state. Specifically, we have obtained two or more intersections in a oscillatory manner, in regions (a1) and (a2) with complex eigenvalues. Whereas, at most two intersections or double QMPE can be found in regions (c), (d) and (e) with EP. Interestingly, even number of intersections (e.g. double QMPE or quadruple QMPE) imply that the initial identities of the two copies (higher valued copy and lower valued copy of an observable, more explicitly hotter and colder copies if the observable is temperature) are restored finally after the last intersection. Therefore, in such cases the initially hotter or higher valued copy remains hotter or higher valued after all the intersections have occurred. On the other hand, odd number of intersections (single QMPE or triple QMPE) indicates the ultimate reversal of the initial identities after the final intersection. In these cases the initially hotter or higher valued copy finally becomes colder or lower valued copy after the completion of all intersections. This classification has important consequences when one is not only interested in faster relaxations or intersections at intermediate times, but also interested in faster reach to final or steady state. It would be intriguing in future to find connections between such odd or even QMPE and faster approach to steady state. Notably, if we restrict on monotonic time volution For that case, even number of QMPE is of no use. Only odd number of QMPE can help to reach faster to steady state using QMPE. However, more conditions are required like avoiding overshoot etc. to use odd QMPE to get faster reach to the steady state. In the next sections, we discuss multiple QMPE in different regimes in detail. In this paper, we illustrate QMPEs in region (d) for quench onto the second-order EP in Sec. IV (with more analytical details in Appendix A) and region (a1) for the quench onto the point with complex eigenvalues in Sec. V (with more analytical details in Appendix B). We do not discuss QMPE in regions (c) and (a2) separately, because the analyses of QMPEs in these regions are similar to those in regions (d) and (a1), respectively. The QMPEs in other regions are discussed in the appendices. Appendix C discusses QMPE in region (b) with purely exponential relaxations and Appendix D contains the description of the QMPE at the third-order EP in region (e).

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Different Regimes

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References IV. QMPE IN REGION (d): SECOND-ORDER EP A. Density matrix elements Using Eqs. (10) and (11) in Eq. (1), we have obtained the following expressions of the density matrix elements ρj (t) (j = 1, 2, 3, 4): where the initial density matrix elements ρn(0) before quench are given by Eq. (5) and the elements ℓk,n are provided in Appendix A. Interestingly, in the presence of the second-order EP in region (d), the density matrix elements in Eq. (13) have picked up an additional algebraic term proportional to time t. This is remarkably different from the time evolution in regions (b) which has only purely exponential relaxations [see Appendix B] and region (a1) that contains damped oscillations [see Sec. V]. The linear t dependence originates from the off-diagonal defect in the Jordan normal form in Eq. (11). To investigate QMPE, we focus on the ground state density matrix element ρgg(t) (≡ ρ4(t)). To detect the QMPE, we would like to find the intersections of the two copies I and II during their relaxation, i.e. solution(s) to the equation ∆ρgg(t) = 0. We obtain the following form of ∆ρgg(t) where the co-efficients α1, α2, α3 are given by The solutions to such transcendental equation in Eq. (14) are given by the Lambert W function, where the equation yey = x has only one solution y = W0(x) if x > 0 and two solutions y = W0(x) and y = W−1(x) if −1/e 6 x < 0. Indeed, such properties of Lambert W function imply that there can be more than one solutions to ∆ρgg(t) = 0, meaning multiple QMPE. More precisely, at most two intersection times i.e. double QMPE can be observed in region (d). Below we provide the criteria for double QMPE and single QMPE in this region: The asymptotic expansions of the functions W0(x) and W−1(x) can be found in Ref. [105]. Note that Lambert W functions have found applications in several fields such as viscous flows [106], crystal growth [107], epitaxial film growth [108], phase separation of polymer mixtures [109], criticality in temporal networks [110] etc. In Fig. 1, we plot the solutions t˜0 and t˜−1 [Eq. (17)] as functions of the driving strengths before quench. Indeed, we observe that there are regions with single QMPE where only t˜0 exists and there are other regions where both solutions prevail leading to double QMPE. In particular, we observe double QMPE for large values of ˜dI and ˜dII. Note that QMPE appears to be unavoidable in the ground state probability ρgg(t) in Fig. 1 i.e. there is no such region in the (d˜ I, d˜ II) where there is no QMPE. B. Energy Here we analyze QMPE in average energy E(t) = Tr[ρb(t)H] where H is the system Hamiltonian presented as a 2 × 2 matrix with elements H11 = 1, H22 = 0, H12 = H21 = ˜d/2 [95]. Consequently, E(t) can be expressed as Note that the average energy directly captures the effect of coherence due to the presence of the off-diagonal density matrix elements in Eq. (18). To explore the QMPE in E(t), we consider the time evolution of the energies starting from the two different initial conditions I and II and look for their intersections during the relaxations. More precisely, we would like to find the solutions to ∆E(t) = 0 where ∆E(t) = EI(t) − EII(t). We obtain the following expression where the elements α1, α2, α3 are given by Eq. (15). Note that Eq. (20) is a similar kind of transcendental equation as Eq. (14), only with different coefficients. Therefore, we can analyze the possibility of QMPE in energy using Lambert W functions as we did for the ground state probability. The two possible solutions to ∆E(t) = 0 are given by where the superscript E denotes energy and is used to differentiate the solutions from Eq. (17) obtained for ground state probability. Figure 2 exhibits the solutions in Eq. (21) as functions of the control parameters ( ˜dI, ˜dII). We observe region with single QMPE (only t˜E 0 exists) as well as region for double QMPE (both t˜E 0 and t˜E −1 exist). In addition, we observe regions where both the solutions are absent i.e. no QMPE. This crucial difference in Fig. 2 compared to Fig. 1 indicates that QMPE in ground state probability does not necessarily implies QMPE in average energy. However, Figs. 1 and 2 also show that there are common parameter values that lead to QMPE both in ρgg(t) and E(t). We present an explicit example in Fig. 3 where both of these quantities exhibit double QMPE, for the same set of parameters. However, the two intersection times for ρgg(t) and E(t) are different from each other. The intersection times for ρgg(t) are comparatively far from each other with the second QMPE being much weaker in magnitude [Fig. 3(a)], whereas the intersection times for E(t) are comparatively much closer to each other, with both the first and second QMPE being comparable in magnitude [Fig. 3(b)]. C. von Neumann entropy Next we consider the von Neumann entropy SvN(t), which is a function of ρb(t), defined as The nonlinearity of SvN(t) as a function of the density matrix elements poses difficulties to perform analytical treatment of the QMPE in von Neumann entropy, in contrast to what we could achieve previously for ground state probability and energy. To observe QMPE in SvN(t), we consider the intersection time(s) for the quantity ∆SvN(t) = S I vN −S II vN. In Fig. 4, we present the behavior of ∆SvN(t) for the same set of parameter values used in Fig. 3. Indeed, we observe that ∆SvN(t) intersects zero twice, implying the existence of double QMPE in the von Neumann entropy, similar to ρgg(t) and E(t). D. Temperature We use the definition of temperature T (t) following Ref. [65] to examine the possibility of thermal QMPE in Hatano’s model, as given below where E(t) and SvN(t) are defined in Eqs. (18) and (22), respectively. The QMPE in system temperature has been observed recently in Ref. [65] using a semi-classical approach. Naturally, it would be interesting to explore thermal QMPE in Hatano’s model that involves the effect of off-diagonal density matrix elements. However, we find that there is no thermal QMPE for the same set of parameters that have produced QMPE for ground state probability [Fig. 3(a)], energy [Fig. 3(b)] and entropy [Fig. 4]. To search for thermal QMPE, we consider a variable dissipation protocol that allows us to use Γ˜ I 6= Γ˜ II 6= Γ˜ (wider parameter space compared to the fixed dissipation protocol Γ˜ I = Γ˜ II = Γ). Indeed, using such protocol, we ˜ have obtained the thermal QMPE shown in Fig. 5 where we present the temperature difference ∆T (t) between two copies I and II. We observe that ∆T (t) becomes zero at some intermediate relaxation time that denotes the intersection of two copies, after which the initially hotter copy becomes colder, leading to thermal QMPE. E. Kullback-Leibler divergence The Kullback-Leibler (KL) divergence is a measure of difference from steady state which is a monotonically decreasing function of time, and it is given by To obtain QMPE in KL divergence, we use the variable dissipation protocol. In Fig. 6(a) we present an example where the copy starting at a higher distance from steady state relaxes faster compared to the other copy with lower initial distance. The copies I and II intersect each other leading to QMPE in DKL(t) and the copy starting at higher initial distance reaches the steady state faster. We have not observed multiple QMPE in KL divergence. It would be an interesting future question to understand if the absence of multiple QMPE in DKL(t) is its generic feature and if the reason is its monotonicity. An important variable of interest in any relaxation process, be it classical or quantum, is the speed of the relaxation. Since the KL divergence gives the distance measure from steady state, a relevant relaxation speed can be defined as where the negative sign is used to complement the fact that the KL divergence decreases with time. It would be interesting to see if the QMPE in DKL(t) is also manifested in the speed vKL(t). For the same set of parameters that produced QMPE in KL divergence [Fig. 6(a)], we investigate QMPE in the speed in Fig. 6(b). Indeed, we observe that the speeds of the two copies I and II intersect each other, leading to QMPE. The higher distant copy goes towards steady state with a higher speed and after the intersection its speed becomes lower. Notably, the intersection times for DKL(t) and vKL(t) are different. In fact, there is a time window where the initially higher distant copy becomes lower distant copy but still maintains its higher speed for some time. It would be intriguing to investigate whether there is some general upper bound on the relaxation speed vKL(t). We end this section by mentioning that the analysis of the QMPE in region (c) would follow similar manner as in region (d) discussed elaborately in the present section. This is because region (c) is also a line of second-order EP just like region (d), the only difference is that the arrangement of eigenvalues for region (c) obeys 0 < λ2 < λ3 = λ4.

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: QMPE in Region

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References II. MODEL AND PROTOCOL We consider a two-level open quantum system, subjected to a periodic drive and dissipatively coupled to the environment. Following Ref. [95], we hereafter refer to this model as Hatano’s model in the present manuscript. In this paper, we adopt ~ = 1. The two levels of the system, ground state and excited state, have energies Eg and Ee, respectively. The gap between the two levels is denoted by ∆ := Ee − Eg. In Hatano’s model, the transfers between the energy levels are enabled by the external oscillatory electric field E = E0cos(ωt) and the dissipation of the system to the environment via coupling Γ. The parameter δ := ∆ −ω represents the detuning of the periodic drive with respect to the system and d := DE0, where D is the electric dipole moment, denotes the effective electric field. The tuning parameters in Hatano’s model are d, Γ and δ . However, one of them fixes the unit of energy, so we essentially have two free parameters ˜d := d/δ and Γ := Γ ˜ /δ. Hatano’s model, within Markovian approximation, can be described by the Gorini-Kossakowski-SudarshanLindblad (GKSL) equation [101–104] (see [95] for the present context):

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Model and Protocol

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References We explore the role of exceptional points and complex eigenvalues on the occurrence of the quantum Mpemba effect. To this end, we study a two-level driven dissipative system subjected to an oscillatory electric field and dissipative coupling with the environment. We find that both exceptional points and complex eigenvalues can lead to multiple quantum Mpemba effect. It occurs in an observable when time evolved copies corresponding to two different initial conditions, one initially having higher observable value compared to the other and both relaxing towards the same steady state, intersect each other more than once during their relaxation process. Each of the intersections denotes a quantum Mpemba effect and marks the reversal of identities between the two copies i.e. the copy with higher observable value before the intersection becomes the lower valued copy (and vice versa) after the intersection. Such multiple intersections originate from additional algebraic time dependence at the exceptional points and due to oscillatory relaxation in the case of complex eigenvalues. We provide analytical results for quantum Mpemba effect in the density matrix in presence of coherence. Depending on the control parameters (drive and dissipation), observables such as energy, von Neumann entropy, temperature etc. exhibit either single or multiple quantum Mpemba effect. However, the distance from steady state measured in terms of the Kullback-Leibler divergence shows only single quantum Mpemba effect although the corresponding speed gives rise to either single or multiple quantum Mpemba effect. I. INTRODUCTION Memory effect describes the influence of past information on present events [1]. Such memory effects often lead to unexpected events. In physical systems, several forms of interesting memory effects have been observed in different contexts e.g. pulse-sign memory in charge-density wave conductors (memory of last input signal direction) [2, 3]; Kaiser effect in rocks and metals [4–6] and Mullins effect in rubbers [7–9] (both bearing memories of largest applied stress); Kovacs effect (memory of bath temperature that affects the thermalization process) [10–14]; aging and rejuvenation in spin glasses [15–17] and structural glasses [18–21], granular materials [22–25], dense suspensions [26] and active matter [27]. The Mpemba effect (MPE) was originally conceived as a counter-intuitive thermal phenomenon of faster cooling of hotter liquids, observed long ago [28] and lately rediscovered by Mpemba and Osborne [29]. The MPE thus bears the imprint of initial memory in a surprising way and constitutes a separate class of fascinating memory effects. Later, the MPE has been regarded as one of the typical anomalous relaxations where a system with initially higher observable (e.g. temperature, energy, entropy, distance function etc.) value can relax faster than its copy with initially lower observable value, when both of them decrease towards the same steady state value. In classical systems, MPE has been found in variety of classical systems including colloids [30, 31], granular fluids [32–38], optical resonators [39–41], inertial suspensions [42, 43], Markovian models [44–47], Langevin system [48] and many others [49–59]. In spite of considerable amount of works on MPE in the classical domain, only a few studies have been done on the quantum Mpemba effect (QMPE), such as quantum Ising model [60], Dicke model [61], few-level systems [62], V-type system [63], XXZ spin chain [64], single level quantum dot [65], XY spin chain [66] and free and interacting integrable systems [67]. In fact, only quite recently the quantum analogue of the original thermal MPE i.e. QMPE in temperature, has been theoretically revealed in a quantum dot [65] and the significant role of relaxation modes other than the slowest relaxation mode to create QMPE, has been shown [65]. However, the semiclassical approach taken in Ref. [65] naturally raises a question about the generality of QMPE in temperature and other observable, in presence of coherence in purely quantum mechanical systems. This constitutes one of the motivations of the present work. Since the phenomenon of QMPE is a comparatively new area of research, it is important to explore the plausible connections of QMPE to other interesting features of quantum systems. In the present work, we would like to address some such connections. In particular, the study of exceptional points (EPs) and surfaces in non-Hermitian quantum systems and Liouvillian dynamics has caught a lot of attention in recent days [68–81] along with their applications in quantum optics [82, 83]. The occurrence of EPs is directly related to eigen-spectrum of a system and hence to its relaxation process. Therefore a natural curiosity would be to understand the role of EPs on the possibility of QMPE. In fact, the occurrence of double MPE (i.e. two intersections between trajectories) at an EP has already been pointed out for MPE in classical inertial suspensions [42]. Interestingly, the acceleration of relaxation process in open quantum systems by maximizing the gap between the zero mode and the slowest eigenmode at an EP, has recently been studied [84]. In general, such speed up is desired for several application purposes and therefore it has been a subject of immense interest in both classical and quantum systems [84–89]. Since QMPE allows faster relaxation of initially higher distant trajectories, it is important to find criteria for QMPE and to understand how to utilize QMPE to achieve faster approach to steady state. Another intriguing feature in non-Hermitian quantum mechanical systems is the appearance of complex eigenvalues [90–94]. It is noteworthy that the analysis of QMPE in the existing examples [60–63, 65] is based on purely exponential relaxation processes, neither the EP nor the oscillations have been considered yet. There is only an exception in Ref. [64] where more than one intersections between trajectories (two intersections to be precise, termed as revivals) has been observed in the entanglement entropy in the XXZ spin chain, although not discussed much and simply explained as a finite-size effect. In this paper, we explore the following two topics: (i) the effect of EPs and complex eigenvalues on QMPE by analysing the Quantum Master Equation, and (ii) the existence of thermal QMPE in a purely quantum mechanical setting. We show that in the presence of EPs and complex eigenvalues, two time-evolved trajectories of an observable (e.g., temperature, energy) can intersect each other more than once before reaching the same final steady state. In the context of thermal QMPE, the first intersection implies that the initially hotter quantum system relaxes faster to intersect the initially colder quantum system and thereby reverses their roles, i.e. hotter becomes colder and vice versa. This can be followed by one or more such intersections so that the present hotter system can become colder again and so on. Similar observations have been obtained in average energy. Such multiple intersections of trajectories give rise to the remarkable phenomenon of multiple QMPE. To demonstrate multiple QMPE as a consequence of EP and complex eigenvalues, we study a two-level driven dissipative quantum system developed in Ref. [95], which we call Hatano’s model in this paper, as a typical and minimal model of open quantum systems. It is well known that many complicated quantum problems are mapped or reduced to two-level systems; needless to mention they are experimentally realizable (e.g., two internal electronic energy levels of an atom controlled by laser) and have applications in qubits, nuclear magnetic resonance, neutrino oscillations, etc. We also find thermal QMPE in presence of coherence, some cases remarkably with multiple intersections, following the same definition of temperature used in Ref. [65] . Importantly, we provide analytical results for the time evolution of the density matrix in the two-level driven dissipative system considered here. Consequently, we derive analytical criteria for QMPE in energy and in principle, one can analyze exactly (analytically and numerically) QMPE in any other observable. The paper is arranged as follows. In Sec. II, we define the details of the model and the protocol used to study QMPE. The Sec. III describes different regions in the control parameter plane depending on the types of eigenvalues of the Lindbladian and the multiple QMPE is defined. In articular, we focus on regions with exceptional points and complex eigenvalues to discuss their effects on QMPE in the density matrix and observables such as energy, entropy, Kullback-Leibler (KL) divergence [96– 100] and temperature; by providing analytical results. In Sec. IV, we discuss QMPE in a region with second-order EPs. Section V demonstrates the effect of complex eigenvalues on the time evolution of the density matrix leading to multiple QMPE in the observables. We summarize the results and discuss possible future directions in Sec. VI. Explicit expressions of the eigenvectors used in the analyses in Secs. IV and V, are provided in Appendix A and Appendix B respectively. We discuss QMPE briefly in region of purely exponential relaxation in Appendix C and at a third-order EP in Appendix D.

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Introduction

While the headline might look a little click-baity, the reason you’re reading this is because you’ve had enough of something Google has done. And… You’re not alone. Google has been slowly making a mess of the internet for a few years now. Maybe not quite as much as the shit show that is Facebook or Meta or whatever the hell they’re called at the moment. But actually, while Zuck’s platforms might be responsible for sucking our attention and (and also proliferating , among other things) - Google sucks for a slightly different and more tangible reason. damaging our mental health hate speech and genocide They’ve basically made all the things that we’ve gotten used to for free online into crappy shadows of what they once were. Sitting comfortably? Let me explain… The Slow Decline of the Search Engine I’m a SEO professional, which means I spend a lot of time looking at and analyzing Google’s search results. I’ve created countless articles for a huge variety of websites, from tech unicorns to niche blogs. And one thing that must be said is that the world's favorite search engine has not been making it easy for either the end-user or those looking to build their platform. To clarify: Around ten years ago, everyone sussed out that keyword stuffing (basically putting their focus keyword tens of times into a blog post) got you onto page one. Cue loads of low-quality content on page one, all of it packed full of obvious keywords. Google decided to fix this, and suddenly there was a strong(er) focus on domain authority. This basically meant stronger domains with lots of backlinks tended to do well. Not always though, and good content was still important, but the backlinks meant you could game the system. Now the SERPs were packed full of crappy websites made to host backlinks with very thin or regurgitated content that didn’t answer the users’ intent. Enter the AI revolution in late 2022. Everyone and their gran is into using ChatGPT to write their shopping list, and also, oh hey… There are blogs packed full of AI content popping up everywhere. Now, I’m not against AI content at all; I think it’s great (if done right). Check my previous article on here: . why AI content isn’t a bad thing But it seems that at Google at least, something clicked. Google’s Helpful Content Update 2023 If you’re in search engine marketing, or you spend any time on Linkedin, you probably noticed everyone was up in arms about the late . 2023 Google algorithm update Now, Google releases probably 3 or 4 big algorithm updates every year. Most result in some ranking changes, usually a little turbulence, and then back to business as usual. Not this puppy. The 2023 edition seems to have royally f**ked a number of quite big publishers, not to mention niche bloggers and site owners. It has also rewarded a number of fairly safe bet forums and magazine sites such as Reddit, Tripadvisor, Forbes, and other big publishers. What’s the reason, many are asking? Well, the general juice is that Google saw that AI content was starting to become a big deal and was starting to dominate the SERPs. So, to combat this, they switched the focus to the ‘safe’ publishers, to the huge detriment of many smaller site owners. OK, you might say. Fair enough. Maybe. Except, if you ask me, much of the time the actual query isn’t being answered - it’s just the trusted (aka big domain authority) sites that have done well from this. Not entirely, I might add. I am currently running a niche blog that has done very well since the helpful update, highlighting that you can still rank well if you do your SEO right. But a quick look at the SERPs shows some more mess. The same algorithm update seems to have placed a bunch of links to Google Books in the search results. Check out this search result below, which is on about page three. Now, you might say that page 3 is irrelevant because, as the , you could hide a dead body on page 2 of Google, and no one would spot it. old joke goes Not so true these days as the first page of Google now scrolls seamlessly into pages two through to pretty much forever. This is one good Google update by the way. But the addition of the book’s link totally messes up the flow and is one of the really sucky things that Google has added in this recent algorithm update. The Hot Mess That Is YouTube Alright, now we come to the exhibit B - which has been a mess for a while now. If you’re old like me, you might remember how YouTube was free and cool, and you could spend hours on there going down cute cat rabbit holes. Google bought them out, and soon enough, ads were popping up every so often. Fine at the time, in moderation. But now it seems that YouTube is basically American TV. Ads pop up halfway through content at inopportune moments, and some of the ad placements are horrible. I’ve seen ads for horror movies during kids’ content - yes, seriously… On top of this, the algorithm that recommends stuff is just… I don’t really know where to start. It’s a mess. I get recommended weird alt-right stuff because I might have watched a click-bait video about the state of the Middle East. And now, I’m being shilled vids about how Jews are responsible for whatever or Muslims are evil. OK. And the search. WT-actual-F? Say you search for a specific music artist on YouTube - maybe the first three to five results will be them (if you’re lucky), and the rest of the content will be random AF. If you ask me, YouTube is barely fit for purpose. But, it’s pretty much the only choice, despite some limp competition from sites like Vimeo and , it seems that YouTube is going to remain the dominant video platform for the foreseeable. DailyMotion Which might be why the youth are turning to TikTok in droves. And Another Thing… It’s not just YouTube; the Google search engine and its algorithms are kinda sucky too by the way. Their paid search platform is a mess and is also a hotbed of . A subject I’ve written about extensively. Yes, they have done some work to try and close this practice down but it’s the gift that keeps giving. ad fraud and click fraud Basically, ad fraud works for Google, so while they’ll be seen to be doing something to fix it, why stop it completely? They make money from it, so hey. So… What’s the Alternative? The thing is, there are alternatives - of course. But by and large, we’re all very stuck in Google’s eco-system. The Google SERPs are still massively important and are far and away the most used method of searching online. Despite reports that TikTok overtook Google a few years back, the fact is, they didn’t. People do tend to use the internet differently now though. We ask Alexa or Siri for local or timely information, we check Instagram or TikTok for some stuff, and we check forums like Reddit for info when we don’t want to sift through the shitty SERPs on Google. Even AI and the rise of Chat GPT have become an alternative for some of us. And while you can search using AI on a number of search engines and browsers, it’s still in its infancy. Until the mainstream susses that AI or social media offer better results, don’t expect things to change any time soon. And while there is always Bing, Google is still going to remain the big dog for a while. Bing’s foray into AI search results has so far not shifted the market much in their favor and probably won’t any time soon. For search engine pros and business managers? It’s time to start looking at strategy, beyond simple search engine rankings. As a content strategist, I focus heavily on SEO and hitting those search engine rankings (e.g., Google). But I’m also very aware of the fact that one algorithm change can wipe out all your organic hard work. Use Google, but be aware: they’re using you too. And make sure to spread your net beyond their eco-system when it comes to marketing.

Why Google Sucks: But We're Stuck With Them (For Now)

কেন গুগল খারাপ: কিন্তু আমরা তাদের সাথে আটকে আছি (এখনের জন্য)

Warum Google scheiße ist: Aber wir bleiben (vorerst) bei ihnen hängen

Por qué Google apesta: pero estamos atrapados con ellos (por ahora)

Pourquoi Google est nul : mais nous sommes coincés avec eux (pour l'instant)

Google बेकार क्यों है: लेकिन हम उनमें फंस गए हैं (अभी के लिए)

Google がダメな理由: しかし、私たちは (今のところ) Google に行き詰まっています

Google이 형편없는 이유: 하지만 우리는 그들과 붙어있습니다(현재로서는)

Por que o Google é uma merda: mas estamos presos a eles (por enquanto)

Почему Google — отстой: но мы застряли на них (пока)

Google Neden Berbat: Ama Biz Onlara Kaldık (Şimdilik)

Tại sao Google tệ: Nhưng chúng tôi đang mắc kẹt với họ (Hiện tại)

为什么谷歌很糟糕：但我们还是被他们困住了（暂时）

Image credit: on Shubham Dhage Unsplash It's no secret that large language models (LLMs) like ChatGPT have transformed how we work today. The trading landscape is no different. Crypto With the introduction of these powerful and new technologies, Crypto traders have thought of various ways they can use these technologies to revolutionize the way traders approach market analysis, generate trading signals, automate trading processes, and even democratize decision-making. In this article, I will mention some ways LLMs can be used to change the way we trade and potentially reshape the trading landscape: Sentiment analysis based trading Strategy LLM models, which are deep learning algorithms, can perform various tasks. One of these tasks is natural language processing (NLP) tasks. This means they can process large volumes of textual data from news articles, social media, and other sources to provide a comprehensive market analysis that can be used for sentiment analysis-based trading. This can save time for traders as they no longer need to sift through large volumes of news sources. So, how does this work? A sentiment analysis-based trading strategy allows traders to gauge the market sentiment of some cryptocurrencies and identify market-moving events. Traders can use the result from this sentiment for market trends analysis, allowing traders to swiftly react to news and decide when to buy and sell. Zanyang talked a bit more about this in his , where he used algorithmic trading platforms like QuantConnect. Use of NLP-Powered sentiment analysis in trading strategy paper Customer support and communication Besides Traders, LLMs can benefit everyone in the crypto space, including cryptocurrency founders, project owners, community leaders, and trading platforms. For community and trading platforms, LLMs can be used for knowledge-based databases, Q&A bots, and customer support. This way, they can seamlessly help users navigate the platform, improve customer experience, answer project-related questions, resolve issues, and assist in technical support. The best thing about this system is that, unlike traditional conversational AI, LLM-powered bots can understand human language and respond better 24/7. Some AI Chatbots you can explore for this are and . AwesomeQA UltimateGPT Generate trading signals A trading signal is a trigger telling a trader when to buy or sell orders. Triggers are generated by simply analyzing indicators, patterns, or data points. These common technical indicators include MACD, RSI, KD, and Bollinger Bands. This talks a lot more about these indicators. article by BYDFi With LLMs, you can generate trading signals in various ways: Building trading algorithms that depend on pre-established rules. Identify trends through sentiment. Interprets news events that can cause market movements. Recognize patterns and interpret news events. With these signals, traders can use the insights to capitalize on anticipated price reactions and make more informed decision-making. The best part is that this signal generation can be automated, thus reducing any latency in trade execution. This exciting discussion on the . Numerai forum discusses using LLMs to create trading signals Build, Build, and Build with LLMs You’ve probably used LLMs in code generation or to assist you in generating codes; If not, I have got an [Analytics Vidhya’s article on How to Build LLMs for Code?](https://www.analyticsvidhya.com/blog/2023/09/llms-for-code/#:~:text=A%20Large%20Language%20Model%20(LLM,understand%20and%20generate%20computer%20code.) for you. Based on your prompt, LLM can be used to write code faster and much more efficiently. You can also use it to generate a whole project. Bringing this idea to the crypto space, you can use LLM models like , , and to generate codes, write smart contracts, and test. These models can also read, analyze and modify your codes or even improve your codebase. WizardCoder StarCoder Codeium All this can be done using simple prompts and knowing what you want. Democratizing decision-making and knowledge One of my favorite things about LLM models is that they make information accessible. You can get information about just about anything by writing a simple prompt. Imagine a model trained on real-world financial data that can understand financial concepts, formulas, and market data. This means you can ask this model about anything in this domain, and it will provide you with information and help you make better decisions. Such a custom LLM would benefit not just market leaders but novice traders and beginners in the crypto space. One of my favorite Crypto knowledge bases is , an AI-powered search engine. MinMax Wrap-up You can use LLMs in various ways, from generating codes to providing 24/7 customer support. While many of these models are still under development, they are becoming increasingly powerful and more accurate as the day passes. As this happens, you will likely use them for even more tasks in the crypto industry in the future.

How LLMs like ChatGPT Can Change the Way We Trade

Before we delve into Part 3 on the evolving landscape of virtual selling, let's quickly recap the series so far. emphasized HubSpot's findings on the pivotal role of platforms like Facebook and TikTok, as well as messaging apps in modern marketing. Building off the first section, , based on Vidyard's comprehensive report, revealed that in-house video production is the new norm, with YouTube as the preferred hosting platform for B2Bs. User-Generated Content (UGC) is dominating the video scene, and analytics reveal short videos garner the most engagement while also offering solid ROI. Part 1 part 2 If you haven't already, I encourage you to review Parts 1 and 2. They provide invaluable insights into the current digital trends and challenges that are shaping the sales and marketing sectors, setting the stage for a careful analysis into virtual selling in this final Part 3. 6 Big takeaways from Vidyard's State of Virtual Selling Report Vidyard's "The State of Virtual Selling" draws insights from over 720 validated sales and marketing professionals. The study sheds light on the challenges and opportunities faced by sales professionals in today's digital-first environment. Here's what I found most interesting: report “63% observed an uptick in response rates, and a third felt that these videos expedited their deal cycles… 41% of users witnessed improved close rates” The Changing Dynamics of Buyer Engagement A majority of sales pros are experiencing increasing difficulty in reaching buyers, with extended sales cycles, more touch points needed to seal deals, and an increase in stakeholders participating in the buying process. This aligns with our earlier findings from HubSpot’s report. "Ghosting" has emerged as a persistent challenge, with 42% of salespeople finding it tough to establish or maintain contact with potential customers. A significant 53% felt that reaching or selling to buyers has become more challenging over the past year. The Power of Custom-Recorded Videos Custom-recorded videos are gaining traction, with a majority agreeing that they outperform text-based emails in terms of opens, clicks, and responses. A staggering 88% of sales professionals reported that custom-recorded videos either matched or surpassed the performance of text-based emails. Interestingly, while text emails tend to induce anxiety in recipients, video emails seem to mitigate this effect. In terms of video styles, 26% were selfie-oriented, 20% showcased screen shares, and a majority (54%) combined both elements (hybrid). The Impact of Video on Sales Metrics Custom-recorded videos have proven beneficial for many sales pros: 63% observed an uptick in response rates, and a third felt that these videos expedited their deal cycles. The influence of custom-recorded videos on close rates is also noteworthy: 41% of users witnessed improved close rates, while a mere 5% experienced a decline. An overwhelming 93% of respondents believe that video content either matches or outperforms other content forms in terms of conversion. The Shift in Sales Cycles and Stakeholder Involvement The sales cycle's duration has seen changes, with 80% of participants stating that deals now take over a month to finalize. For half of the respondents, the sales cycle now spans between 4 to 18 months. The number of touch points in the sales process has surged for 57% of sales professionals. The buying process is increasingly collaborative, with a mere 7% selling to a single decision-maker. The majority navigate through 2-to-4 person buying committees, either intra-departmental (30%) or inter-departmental (25%). A significant 40% of sales pros have observed an increase in the number of stakeholders involved in the sales process over the past year. Video Usage and Adoption Trends The adoption of video in business has skyrocketed, marking a 135% increase year-over-year. On average, businesses are now producing 510 new videos annually, marking a 49% surge from the previous year. Vidyard's report underscores the pivotal role of custom-recorded videos in reshaping the virtual selling landscape. As businesses navigate the challenges of extended sales cycles and increased stakeholder involvement, the strategic use of video emerges as a powerful, low-cost and increasingly efficient tool to enhance engagement, response rates, and overall sales outcomes. Industries are increasing their video usage universally, but I'd like to particularly highlight the real estate sector and its swift and effective adaptation. The insights from this industry will further validate the aforementioned trends and provide research on effective strategies, their applications, and the key performance indicators that confirm their return on investment. 5 Insights Into Real Estate's Digital Evolution: Embracing Video for Enhanced Engagement The real estate industry, which traditionally depended on face-to-face interactions, has undergone a digital transformation, with video content taking center stage. Vidyard's analysis provides valuable insights into how real estate professionals are using video to revolutionize their methods, offering lessons for other sectors to learn from their achievements. [video messages are] 40% more likely to be opened, 37% more likely to be clicked on, and have a response rate that's three times higher than non-video content. Surge in Video Adoption The real estate industry has experienced a remarkable 132% uptick in video creation year-over-year. Furthermore, there's been a 155% increase in user-generated videos compared to the previous year. This surge underscores the sector's recognition of video as an indispensable tool for engagement. Why Video is Reshaping Real Estate Several compelling reasons account for real estate's rapid pivot to video: Clarity in Communication: Videos simplify complex explanations, making property features and contractual details more accessible to potential buyers. Enhanced Engagement: Video messages, whether shared via email or social media, boast impressive engagement metrics. They're 40% more likely to be opened, 37% more likely to be clicked on, and have a response rate that's three times higher than non-video content. Versatility and Customization: Once created, videos can be easily repurposed, reshared, and tailored, ensuring each communication feels personalized. Efficiency and Connection: Videos expedite deal progress by swiftly addressing queries, highlighting property amenities, and fostering a genuine personal connection between agents and clients. The Power of Cold Outreach Videos Real estate agents who venture into cold outreach video messages are reaping dividends. These videos not only capture attention but also leverage social media algorithms. For instance, Instagram videos garner 38% more engagement compared to non-video posts, while Facebook's video engagement soars to an impressive 135% versus non-video posts. Keeping Deals Alive with Video In the competitive real estate market, keeping potential deals warm is crucial. Agents found that follow-up videos outperform traditional emails in this regard. The perceived effort behind creating a video translates to higher value in the eyes of clients, helping to ensure sustained interest. Showcasing Personality for Client Acquisition For real estate agents, personality can be a significant selling point. Videos offer a platform where agents can let their unique personalities shine, providing both current and potential clients with a blend of personal and, at times, entertaining experiences. This approach not only helps fosters client loyalty but also helps earn more referrals, ensuring a steady stream of satisfied home buyers and sellers. Conclusion These reports show us how influential video, socials, UGC and AI have become in today's online world. We also see that consumer behaviour is changing and how businesses are adapting. Sales pros are finding that old tactics are increasingly ineffective at closing deals and earning new business - and it’s forcing industries like real estate to get creative to stay competitive. As technology and what people want keep changing, businesses need to keep up and attempt to predict what might be next. The findings from these reports may leave us asking: How will AI and video change the way people connect with brands in the next few years? With so many changes online, how can businesses make sure they stand out and give people what they really want? Just like real estate is using more videos now, which other businesses might start doing the same? In a time when we talk to each other more through screens, how can businesses keep things real and honest? And with more and more videos popping up because AI makes it easier to scale, will people get tired of them? Will they keep evolving to get shorter and more punchy? And if not that, what might be the next big trend(s) with video? As we navigate this dynamic digital terrain, understanding these key questions is not just insightful—it's essential for strategic planning. Grasping the future roles of AI, video, and shifting consumer behavior will not only redefine brand engagement but also identify the trailblazers in the next wave of online business. In a world where technology is constantly reshaping the sales and marketing landscape, those who adapt, innovate, and anticipate will not merely survive—they will set the industry standard. If you found this article insightful, please consider sharing it, leaving a comment, or tagging me on , , or where this discussion can continue to evolve. Let's lead the way together in this exciting digital frontier! HackerNoon LinkedIn X

Digital Mastery 2023, Part 3: The Evolving Landscape of Virtual Selling

ডিজিটাল মাস্টারি 2023, পার্ট 3: ভার্চুয়াল সেলিংয়ের বিবর্তিত ল্যান্ডস্কেপ

Digital Mastery 2023, Teil 3: Die sich entwickelnde Landschaft des virtuellen Verkaufs

Dominio digital 2023, parte 3: El panorama en evolución de la venta virtual

Maîtrise numérique 2023, partie 3 : le paysage en évolution de la vente virtuelle

डिजिटल महारत 2023, भाग 3: वर्चुअल सेलिंग का विकसित परिदृश्य

Digital Mastery 2023、パート 3: バーチャル販売の進化する風景

2023년 디지털 숙달, 3부: 진화하는 가상 판매 환경

Digital Mastery 2023, Parte 3: O cenário em evolução das vendas virtuais

Цифровое мастерство 2023, часть 3: развивающаяся ситуация в сфере виртуальных продаж

Dijital Ustalık 2023, Bölüm 3: Sanal Satışın Gelişen Ortamı

Làm chủ kỹ thuật số 2023, Phần 3: Bối cảnh phát triển của hoạt động bán hàng ảo

掌握数字化 2023，第 3 部分：虚拟销售不断发展的格局

In , we reviewed key findings from HubSpot's "State of Marketing 2023" report. Facebook remains a top marketing tool, short videos formats like TikTok’s are rising, and messaging apps are becoming crucial for business. Sales professionals are also facing growing challenges in prospecting, engaging and closing deals. We also highlighted the growing role of AI in marketing. Part 1 This sets us up for Part 2, focusing on the specific impact of video in business in 2023. Highlights from Vidyard's Video In Business Benchmark Report: 13 Changes and trends to know Vidyard’s Video In Business Benchmark Report unique data set comes from 1,778,000 videos, anonymized from their customers. Findings both align with our HubSpot’s report highlights, and yield additional insight into video trends, analytics, engagement metrics, and the most popular formats in use. Tuesday stands out as the most popular day for releasing videos, while Wednesday garners the highest video view Video Production and Types Diverse Video Portfolio: The leading types of videos produced by businesses are webinars (42%), demos (21%), and social media videos (15%). A significant 92% of businesses now produce videos using internal resources, with the average production cost ranging between $1,000 to $5,000. In-house Production: 52% of all businesses that responded have now taken their video production in-house. Video Investment Types 87% of businesses use screen recording tools. 47% of companies have invested in product demo videos, more than any other type of produced video. 35% have invested in explainer videos, 33% of companies have invested in recorded webinars videos, 31% How-to videos and 27% Customer testimonial videos. Video Platforms and Publishing YouTube is the top platform for hosting videos, used by 92% of businesses. Vidyard found that customer hosted videos on their websites at (84%) and post them on LinkedIn (70%). Platform Preferences: Tuesday stands out as the most popular day for releasing videos, while Wednesday garners the highest video views. Publishing Patterns: A remarkable 72% of videos produced this year were user-generated, with the majority being screen recordings or hybrid videos (80%) like the image pictured below. User-Generated Content (UGC) UGC Dominance: 54% of sales pros say sending hybrid videos to buyers gets the most responses. 74% of outbound sales professionals leverage UGC videos to enhance their sales outcomes, with hybrid videos being particularly effective. Sales and UGC: Video plays a pivotal role in sales, with 71% of businesses now leveraging it in their sales dialogues. 54% of sales pros say sending hybrid videos to buyers gets the most responses. Video in Sales: The final two sections of this article will go deeper into what’s working the best. The average video length is 4 minutes and 20 seconds, a decrease from the previous 6-minute average. Video Metrics Analytics and Video Duration: 60% of all videos are under 2 minutes, the recommended size to maximize retention. Another trend of interest; videos under 60 seconds, which make up 16% of all videos, are gaining traction. This lines up with our earlier HubSpot findings on short form video’s meteoric rise and rise. Videos under 60 seconds boast the highest engagement, retaining an average of 68% of viewers. Engagement Metrics: In contrast, videos between 2-4 minutes achieve the highest completion rate of 52%. A significant 85% of businesses utilize video analytics, focusing on metrics like play rate (78%), engagement (63%), and conversion rate (35%). Video Analytics: Blog posts incorporating video attract 3x as many as blog posts without video. inbound links While 30% of businesses plan to ramp up their video budget, a robust 89% believe that their video initiatives offer a solid ROI. Investment and ROI Video Budgeting Video Effectiveness Including a video on a landing page can increase conversion by 80%. Landing Pages: Incorporating video in emails can skyrocket the click-through rate by 200-300%. That’s big news when benchmark email metrics on average are declining, as mentioned earlier. Email Marketing: Blog posts incorporating video attract 3x as many inbound links as blog posts without video. Video Blog Posts: As mentioned in earlier writing, blog posts that contain a video have a 53X higher chance of ranking on the first page of Google A resounding 94% of businesses agreed that video is enhancing user comprehension of their offerings. Video Impact and Adoption Video's Influence: The primary hurdles in video production are time constraints (24%), budget limitations (23%), and resource scarcity (22%). Video Production Challenges: Not surprisingly, a substantial 70% of businesses are gearing up to amplify their video marketing efforts, and 96% have already integrated video into their marketing blueprint. Video in Marketing: An overwhelming 99% of businesses will continue to use video in 2023, with 96% planning to either ramp up or maintain their current spending. Future Outlook: As we wrap up Part 2 of the Digital Mastery 2023 Series, it's clear that video is no longer just an option—it's a necessity for businesses aiming to stay ahead. From in-house production to leveraging User-Generated Content, video is revolutionizing how we engage, sell, and measure success. With a staggering 99% of businesses planning to continue or increase their video usage, the medium is undeniably shaping the future of digital marketing and sales. If you found this deep dive into the new benchmarks for video in business insightful, share the it with your friends, co-workers and network, leave a comment, , , or . Stay tuned for Part 3, where we'll explore the evolving landscape of virtual selling in this digital age. follow me on HackerNoon LinkedIn X READ NOW: Digital Mastery 2023, Part 3: The Evolving Landscape of Virtual Selling

Digital Mastery 2023, Part 2: The New Benchmarks For Video In Business

ডিজিটাল মাস্টারি 2023, পার্ট 2: ব্যবসায় ভিডিওর জন্য নতুন বেঞ্চমার্ক

Digital Mastery 2023, Teil 2: Die neuen Maßstäbe für Videos in der Wirtschaft

Digital Mastery 2023, Parte 2: Los nuevos puntos de referencia para el vídeo en los negocios

Digital Mastery 2023, Partie 2 : Les nouveaux benchmarks de la vidéo en entreprise

डिजिटल महारत 2023, भाग 2: व्यवसाय में वीडियो के लिए नए मानक

Digital Mastery 2023、パート 2: ビジネスにおけるビデオの新しいベンチマーク

2023년 디지털 숙달, 2부: 비즈니스 동영상의 새로운 벤치마크

Digital Mastery 2023, Parte 2: Os novos benchmarks para vídeo nos negócios

Dijital Ustalık 2023, Bölüm 2: İş Dünyasında Video İçin Yeni Kriterler

Làm chủ kỹ thuật số năm 2023, Phần 2: Các tiêu chuẩn mới cho video trong kinh doanh

数字掌握 2023，第 2 部分：商业视频的新基准

The Challenges, Costs, and Considerations of Building or Fine-Tuning an LLM

এলএলএম নির্মাণ বা ফাইন-টিউনিংয়ের চ্যালেঞ্জ, খরচ এবং বিবেচনা

Die Herausforderungen, Kosten und Überlegungen beim Aufbau oder der Feinabstimmung eines LLM

Los desafíos, costos y consideraciones de crear o perfeccionar un LLM

Les défis, les coûts et les considérations liées à la création ou à la mise au point d'un LLM

एलएलएम के निर्माण या फाइन-ट्यूनिंग की चुनौतियाँ, लागत और विचार

LLM の構築または微調整における課題、コスト、考慮事項

LLM 구축 또는 미세 조정에 대한 과제, 비용 및 고려 사항

Os desafios, custos e considerações de construir ou ajustar um LLM

Yüksek Lisans (LLM) Oluşturmanın veya İnce Ayar Yapmanın Zorlukları, Maliyetleri ve Dikkat Edilmesi Gereken Hususlar

Những thách thức, chi phí và những cân nhắc khi xây dựng hoặc tinh chỉnh một LLM

建立或微调法学硕士的挑战、成本和注意事项

In an era marked by rapid digital transformation and evolving consumer behaviours, staying on top of industry trends is key for sales, marketing, and leadership professionals. Knowing what trends have sprouted today will give businesses a head start on where tomorrow’s big growth opportunities will be found. Today, I've condensed over 20 hours of research into three bite sized articles. So at the end, and if you found it insightful, please share with your network! In particular, we will review key highlights from three different sales and marketing reports, along with notable industry analysis that will provide you with clues on tomorrow’s digital landscape best-practices. HubSpot's "State of Marketing 2023" unveils the latest shifts in digital marketing, emphasizing the continued dominance of social media and the challenges faced by sales teams. Complementing this, Vidyard's "Video in Business Benchmark " and "The State of Sales and Virtual Selling " offer a deep dive into the transformative role of video in business communication, sales, and marketing. To close, I chose Vidyard's on the real estate sector, revealing how agents are pioneering video strategies to enhance engagement, foster genuine connections, and drive results. Together, these insights provide a comprehensive roadmap for professionals keen on harnessing the power of digital trends to elevate their strategies, outcomes, and profits. report Report Report customer research If this all sounds good, then let’s jump right into it! 13 Trends That Stand Out From HubSpot State of Marketing Report 2023 HubSpot’s State of Marketing Report for 2023 survey includes 1,200+ B2B and B2C marketing professionals and the company has been a great resource over the past few years - one I often come back to for sharpening skills and updates. 34% of marketers selected “paid advertising” as the most overstated marketing tactic Despite the emergence of new platforms, Facebook remains the most leveraged social media platform, with 64% of marketers utilizing it in 2022. Instagram (58%), YouTube (57%), and X/Twitter follow closely (43%), highlighting the continued importance of these platforms in marketing strategies. Interestingly, TikTok, a relatively newer platform, has quickly risen to prominence, used by 42% of marketers and LinkedIn (33%). Social Media Dominance When it comes to ROI, Facebook leads the pack at 22%. However, platforms like YouTube, Instagram, and TikTok are not far behind, each offering a 16% ROI. This underscores the growing importance of visual content, particularly short-form videos, in driving engagement and conversions. We’ll discuss this in much more detail in the next 3 sections. Return on Investment (ROI) HubSpot users observed several significant shifts were in 2022. Changing Benchmarks: Web traffic saw a decline of 6% compared to 2021. Email open rates dropped by 12%, indicating potential email fatigue among consumers. Growing use of AI in email marketing will no doubt increase this negative trend for many in 2023. However we’ll go over how smart marketers are increasing email open rates with video in the next section. On a positive note from the benchmark report, HubSpot’s users saw web conversions increased by 11%, and inbound leads grew by 6.66%, suggesting that while traffic might be decreasing, the quality of leads is improving. Mobile marketing doesn’t just mean a mobile friendly website anymore, but mobile messaging with SMS, Messenger, WhatsApp and other platforms saw 40% of respondents emphasizing their effectiveness. It's clear that mobile-first strategies are no longer optional but essential. Mobile Marketing With 71% of consumers open to using messaging apps for customer assistance and 24% of marketers integrating them into their marketing plans, it's evident that direct communication channels are gaining traction. The Power of Messaging Apps Salespeople face increasing challenges, from getting responses from prospects (40%) to securing appointments (38%) and closing deals is getting harder (30%). Engaging multiple decision-makers has increased (28%) and capturing attention are also growing concerns, highlighting the need for more personalized and targeted marketing and sales strategies. Challenges in Sales Short-form video content ROI, such as TikTok, Reels, and YouTube shorts, has emerged as the top marketing trend, according to a whopping 33% of respondents. Of those already doing shoft-form content, 90% of these marketers are planning to increase or maintain their investment in this format, it's evident that short-form video is here to stay. The Rise & Rise of Short-form Video Content Video continues to reign supreme as the most popular and effective media format for the fourth consecutive year. However, other content forms like images, blogs, case studies, customer testimonials, and podcasts also play a crucial role in a holistic marketing strategy. Content is Still King The future looks optimistic, with 92% of marketers expecting their budgets to either remain consistent or increase in 2023. Budgeting and Investment 34% of marketers selected “paid advertising” as the most overstated marketing tactic and 28% have already began reducing their advertising budget to fund more content marketing. Content remains a pivotal tool for businesses, with findings such as: of marketers say video content is the most important. said infographics were, followed by saying ebooks and whitepapers. Content Marketing Insights 16% 14% 3% of marketers support 2 to 4 roles and buyer personas with dedicated content. 52% of buyers viewed between 3 to 5 pieces of content before engaging with a sales rep. 47% of B2B marketers use content marketing strategies. 89% of B2B buyers are now seeking out content with input from industry leaders. 96% of top businesses are using content marketing to drive sales. 91% of marketers now use interactive content. Examples include calculators, quizzes, ebooks, videos, and animated infographics. 33% of CMOs view custom content as the future of marketing. 78% of highly effective organizations have someone in charge of content strategy. 86% of businesses worldwide intend to spend more on content creation. 56% of organizations are now using inbound marketing as their primary approach. 73% A significant of marketers lacking a consistent content strategy, indicating a potential area for improvement. 70% SEO remains a top priority for 61% of marketers, emphasizing its continued relevance. SEO 47% of marketers stated that they are planning to use SEO in their marketing efforts. Social Media of marketers use social media for market research. 76% of people say they use Facebook for professional purposes. 74% of marketers are adding YouTube to their marketing plan. 48% of marketers are adding Instagram to their marketing plan. 39% of marketers are adding Facebook video to their marketing plan in 2023. 46% Platforms like LinkedIn are growing in prominence in B2B marketing, with 45% of marketers identifying it as the best network for their needs. While the report touches upon various aspects of sales and marketing, the underlying theme is the growing influence of AI. From content creation to predictive analytics, AI's potential to revolutionize the industry is undeniable. The Role of AI in Marketing Last week I shared which is worth reading further, for those interested. 50 AI stats that highlight marketing’s transformation So as businesses continue to integrate AI-driven tools and strategies, the future of marketing looks more personalized, efficient, interactive and data-driven. Earlier this month I documented that make good on the promise that AI is already giving marketers more hours in the day. 28 innovative AI marketing strategies and applications Next, we’ll go into more detail about how video is changing sales and marketing across the board in 2023 and beyond. Digital Mastery 2023, Part 2: New Benchmarks For Video In Business

Digital Mastery 2023, Part 1: Shifting Preferences and Behaviour Online

ডিজিটাল মাস্টারি 2023, পার্ট 1: অনলাইনে পছন্দ এবং আচরণ পরিবর্তন করা

Digital Mastery 2023, Teil 1: Präferenzen und Verhalten im Internet ändern

Dominio digital 2023, parte 1: Cambio de preferencias y comportamiento en línea

Maîtrise numérique 2023, partie 1 : changements de préférences et de comportements en ligne

डिजिटल महारत 2023, भाग 1: प्राथमिकताएँ और व्यवहार ऑनलाइन बदलना

Digital Mastery 2023、パート 1: 好みと行動をオンラインに変える

2023년 디지털 숙달, 1부: 온라인 선호도 및 행동 변화

Digital Mastery 2023, Parte 1: Mudança de preferências e comportamento online

Dijital Ustalık 2023, Bölüm 1: Çevrimiçi Tercihler ve Davranışların Değişmesi

Làm chủ kỹ thuật số 2023, Phần 1: Thay đổi sở thích và hành vi trực tuyến

掌握数字化 2023，第 1 部分：改变在线偏好和行为

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