Indirect detection of Cosmological Constant from N entangled open quantum system
Subhashish Banerjee, Sayantan Choudhury, Satyaki Chowdhury, Rathindra Nath Das, Nitin Gupta, Sudhakar Panda, Abinash Swain
IIndirect detection of Cosmological Constant from large N entangledopen quantum system Subhashish Banerjee , Sayantan Choudhury , ∗ ,Satyaki Chowdhury , , Rathindra Nath Das , Nitin Gupta , Sudhakar Panda , , Abinash Swain , ∗ Indian Institute of Technology Jodhpur, Jodhpur 342011, India. Quantum Gravity and Unified Theory and Theoretical Cosmology Group,Max Planck Institute for Gravitational Physics (Albert Einstein Institute),Am M ¨ u hlenberg 1, 14476 Potsdam-Golm, Germany. National Institute of Science Education and Research, Jatni, Bhubaneswar, Odisha - 752050, India. Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai - 400085, India. Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai - 400076, India. and Department of Physical Sciences, Indian Institute of Science Education & Research Mohali, Manauli PO 140306 Punjab India.
IIn this letter, we study the indirect detection of
Cosmological Constant from an open quantumsystem of N entangled spins, weakly interacting with a thermal bath, a massless scalar field min-imally coupled with the static De Sitter background, by computing the spectroscopic shifts. Byassuming pairwise entanglement between spins, we construct entangled N states using a generali-sation of the superposition principle. We have found that in the realistic large N limit, where thesystem consists of N ∼ O (10 − ) spins, the corresponding spectroscopic shifts, caused by theeffective Hamiltonian of the system due to Casimir Polder interaction with the bath, play a crucialrole to determine the observationally consistent Cosmological Constant, Λ ∼ O (10 − ) (Planckianunits) in the static patch of De Sitter space. In recent times the study of the quantum systemsthat are interacting with their surroundings has acquireda lot of attention in different fields ranging from con-densed matter [1–4], quantum information [5], subatomicphysics [6–11], quantum dissipative systems [12], holog-raphy [13, 14] to cosmology [5, 15–46] for a sample of therelevant literature. Here our interest is the study of thecurvature of the static patch of De Sitter space as well asthe Cosmological Constant from the spectroscopic Lambshift [47–49]. The system under consideration is an openquantum system of N entangled spins which are weaklycoupled to their environment, modelled by a masslessscalar field minimally coupled to static patch of De Sitterspace-time. We are interested to study how the entangledstates of the system and the Lamb shift change affect thecurvature of the static patch of De Sitter space-time aswell as the Cosmological Constant as the number of spinsbecome very large in the thermodynamic limit, in real-istic physical situations. One can design such a thoughtexperimental condensed matter analogue gravity [50, 51]set up of measuring spectroscopic shift in an open quan-tum system in a quantum laboratory to get a proper es-timation of the curvature of the static patch of De Sitterspace as well as the Cosmological Constant without re-course to any cosmological observation. This is the mainhighlight of this letter, where our claim is that, with-out doing any cosmological observation one can measurethe value of the Cosmological Constant from quantumspectroscopy of open systems. For large N spin system,where the number of spins, N ∼ O (10 − ), we showfrom our analysis that the obtained value of the Cosmo- logical Constant is perfectly consistent with the presentday observed central value of the Cosmological Constant,Λ observed ∼ . × − in the Planckian unit [52].The open quantum set up can be described by thefollowing Hamiltonian: H T = H S ⊗ I + I ⊗ H B + H I , (1)where H S , H B and H I respectively describes the Hamilto-nian of the spin system, bath and the interaction betweenthem. Also I and I are the (2 ×
2) identity opera-tors for the system and bath, respectively. We choose ourspin Hamiltonian in such a way that the individual Paulimatrices are oriented arbitrarily in space. In the presentcontext, the N spin system Hamiltonian is described by: H S = ω N (cid:88) δ =1 3 (cid:88) i =1 n δi .σ δi , (2)where n δi represent the unit vectors along any arbitrary i (= 1 , , δ = 1 , · · · , N . Also, σ δi ,( i = 1 , , δ .The free rescaled scalar field, minimally coupled with thestatic De Sitter background is considered as the bath, andis described by the following Hamiltonian: H B = (cid:90) ∞ dr (cid:90) π dθ (cid:90) π dφ (cid:2) Π / r sin θ/ (cid:110) r ( ∂ r Φ) + (cid:0) − r /α (cid:1) − (cid:0) ( ∂ θ Φ) + ( ∂ φ Φ) / sin θ (cid:1)(cid:111)(cid:105) . (3)Here, Π Φ represents the momentum canonically conju-gate to the scalar field Φ( x ) in the static De Sitter patch. a r X i v : . [ h e p - t h ] M a y As a choice of background classical geometry we haveconsidered here the static De Sitter patch as our primeobjective to implement the present methodology to thereal world cosmological observation. The static De Sittermetric (which we will define later) contains the Cosmo-logical Constant term explicitly which is one of the primemeasurable quantities at late time scale (mostly at thepresent day) in Cosmology. Using this analogue gravitythought experiment performed with N spins our objec-tive is to measure the value of Cosmological Constantat present day from the spectroscopic shift formula indi-rectly. For this purpose we have only taken the observedvalue of Cosmological Constant to check the consistencyof our finding from this methodology. Not only the nu-merical value of the Cosmological Constant, but also thecurvature of static patch of De Sitter space can be furtherconstrained using the present methodology. The interac-tion between the N spin system and the thermal bathplays a crucial role in the dynamics of open quantumsystem. For the model being considered, the interactionbetween the system of N entangled spins and the bath isgiven by: H I = µ N (cid:88) δ =1 3 (cid:88) i =1 ( n δi .σ δi )Φ( x δ ) , (4)where the parameter µ represents the coupling betweenthe system and the bath and is taken to be sufficientlysmall. Also, it is important to note that in the interac-tion Hamiltonian we have restricted upto quadratic con-tribution. Any higher order non-linear interactions areavoided for the sake of simplicity, but for a generalisedcase one can include such contributions in the presentanalysis.The normalized N spin entangled states for the systemHamiltonian are given by: | G (cid:105) ∝ N (cid:88) δ,η =1 ,δ<η | g δ (cid:105) ⊗ | g η (cid:105) , | E (cid:105) ∝ N (cid:88) δ,η =1 ,δ<η | e δ (cid:105) ⊗ | e η (cid:105)| S (cid:105) , | A (cid:105) ∝ N (cid:88) δ,η =1 ,δ<η ( | e δ (cid:105) ⊗ | g η (cid:105) ± | g δ (cid:105) ⊗ | e η (cid:105) ) / √ , (5)where | g δ (cid:105) , | e η (cid:105)∀ δ, η = 1 , · · · , N are the eigen vectors forindividual atom corresponding to ground (lower energy)state and excited (higher energy) state. Here we alsodefine the proportionality constant of the normalizationfactor as, N norm = 1 / (cid:112) N C = (cid:112) N − /N !. Thenormalization constant has been fixed by taking the innerproducts between elements of the direct product spacewith the restriction that the inner product only acts be-tween elements belonging to the same Hilbert space ofthe open quantum system under consideration.At the starting point we assume separable initial con-ditions, i.e., the total density matrix ρ T at the initialtime scale τ = τ factorizes as, ρ T ( τ ) = ρ S ( τ ) ⊗ ρ B ( τ ) , where ρ S ( τ ) and ρ B ( τ ) constitute the system and bathdensity matrices at initial time τ = τ , respectively. Asthe system evolves with time, it starts interacting withits surrounding which we have treated as a thermal bathmodelled by massless scalar field placed in the static DeSitter background. Since we are interested in the dy-namics of our system of interest (sub system), made bythe N spins, we consider its reduced density matrix bytaking partial trace over the thermal bath, i.e., ρ S ( τ ) =Tr B [ ρ T ( τ )]. Though the total system plus bath joint evo-lution is unitary, the reduced dynamics of the system ofinterest is not. The non-unitary dissipative time evolu-tion of the reduced density matrix of the sub system inthe weak coupling limit can be described by the GKSL(Gorini Kossakowski Sudarshan Lindblad) master equa-tion [15], ∂ τ ρ S ( τ ) = − i [ H eff , ρ S ( τ )] + L [ ρ S ( τ )], where L [ ρ S ( τ )] is the Lindbladian operator which captures theeffects of quantum dissipation and non-unitarity. Theeffective Hamiltonian, for the present model, is H eff = H S + H LS , where H LS ( τ ) is the Lamb shift Hamiltoniangiven by: H LS = − i N (cid:88) δ,η =1 3 (cid:88) i,j =1 H ( δη ) ij ( n δi .σ δi )( n ηj .σ ηj ) . (6)We consider interaction between two spins at a time,which can be implemented in terms of the Pauli operatorsas, σ δi = σ i ⊗ I ,B (for even & odd δ ), σ δi = I ,S ⊗ σ i (foreven & odd δ ) and σ δi = I ,S ⊗ I ,B (for odd δ ), wheremathematical structure of, I ,S and I ,B are identical.In the Lamb shift the time dependent coefficient matrix H ( δη ) ij ( τ ) can be obtained from the Hilbert transform ofthe N spin Wightman function, which is computed inthe static De Sitter patch, described by the following 4Dinfinitesimal line element: ds = (cid:0) − r /α (cid:1) dt − (cid:0) − r /α (cid:1) − dr − r d Ω . (7)Here, the parameter α = (cid:112) / Λ, where Λ > N spin Wightman functions, which arebasically two point functions in quantum field theory atfinite temperature. Consequently, the diagonal entries(auto-correlations) of the N spin Wightman function arecalculated as: G αα ( x, x (cid:48) ) = G ββ ( x, x (cid:48) ) = − / (16 π k sinh f (∆ τ, k )) , (8)where we define, f (∆ τ, k ) = (∆ τ / k − i(cid:15) ) and (cid:15) is aninfinitesimal contour deformation parameter. Also theoff-diagonal (cross-correlation) components of the N spinWightman function can be computed as: G αβ ( x, x (cid:48) ) = G βα ( x, x (cid:48) ) = − (16 π k ) − (cid:8) sinh f (∆ τ, k ) − r k sin (cid:0) ∆ θ (cid:1)(cid:9) . (9)Here the parameter k can be expressed as, k = √ g α = √ α − r = (cid:112) / Λ − r >
0. Further, the curvature ofthe static De Sitter patch can be expressed in terms of theRicci scalar term, given by, R = 12 /α . This directly im-plies that one can probe the Cosmological Constant fromthe static De Sitter patch using the spectroscopic shift.The shifts for identical N entangled spins can be phys-ically interpreted as the energy shift obtained for eachindividual spin immersed in a thermal bath, described bythe temperature, T = 1 /β = 1 / πk = (cid:112) T + T , (with Planck’s constant (cid:126) = 1 and Boltzmann con-stant k B = 1) where the Gibbons-Hawking and
Unruh temperature are defined as, T GH = 1 / πα, T Unruh = a/ π, with a = ( r/α ) (cid:0) − r /α (cid:1) − / . When spinsare localised at r = 0, then a = 0, which in turn implies, T = T GH . Here the temperature of the bath T can alsobe interpreted as the equilibrium temperature which canbe obtained by solving the GKSL master equation forthe thermal density matrix in the large time limit. Ini-tially when the non-unitary system evolves with time itgoes out-of-equilibrium and if we wait for long enoughtime, it is expected that the system will reach thermalequilibrium. The N dependency comes in the states, inthe matrix H δηij and the direction cosines of the align-ment of each spin. The generic Lamb shifts are given by, δE Ψ = (cid:104) Ψ | H LS | Ψ (cid:105) , where | Ψ (cid:105) is any possible entangledstate. Here the spectral shifts for the N spins derived as: δE NY N DC = δE NS Γ N DC = − δE NA Γ N DC = −F ( L, k, ω ) / N , (10)where Y represents the ground and the excited statesand S and A symmetric and antisymmetric states, re-spectively. Here, Γ Ni ; DC ∀ i = 1 , , N number of identical spins. These an-gular factors become extremely complicated to write forany arbitrary number of N spins. Because of this factit is also expected that as we approach the large N limit we get extremely complicated expressions. For allthe spectral shifts we get an overall common factor of N − = N C = N ! / N − F ( L, k, ω ), givenby, F ( L, k, ω ) = E ( L, k ) cos (cid:0) ω k sinh − ( L/ k ) (cid:1) , (11)where, E ( L, k ) = µ / (8 πL (cid:112) L/ k ) ). In this con-text, L represents the euclidean distance between anypair of spins, and is L = 2 r sin(∆ θ/ θ rep-resents the angular separation, which we have assumedto be the same for all the spins. In different euclideanlength scales, we have: F ( L, k, ω ) = (cid:40) µ k πL cos (2 ω k ln ( L/ k )) , L >> kµ πL cos ( ω L ) . L << k (12) Here, P is the principal part of the Hilbert transformed integral of N point Wightman function. For a realisticsituation we take the large N limit, using the Stirling-Gosper approximation, as a result of which the normal-ization factor can be written as: N norm Large N −−−−−→ (cid:92) N norm ≈ √ (cid:32) − (cid:0) N + (cid:1) (cid:33) / (cid:18) Ne (cid:19) − N/ (cid:18) N − e (cid:19) N/ − (cid:118)(cid:117)(cid:117)(cid:116) − ( N + ) (cid:0) − N (cid:1) . (13)Here we use, N ! ∼ (cid:112) (2 N + 1 / π ( N/e ) N (1+(1 / N )).Thus shifts can be approximately derived as : (cid:91) δE NY N DC = (cid:91) δE NS Γ N DC = − (cid:91) δE NA Γ N DC = −F ( L, k, ω ) / (cid:92) N norm2 , (14)In the large N limit, behaviour of F ( L, k, ω ) remains FIG. 1. Behaviour of the spectroscopic shifts with the numberof entangled spins. Here we fix µ = 0 . L = 10 and ω = 1for the given value of the curvature R = 1 . unchanged, as the euclidean distance L , inverse of thecurvature parameter k and the frequency ω of the N number of identical spins are not controlled by N . Also,for large N the normalization factor asymptotically sat-urates to √ / N ). In fig. (1), the behaviour ofshifts with the number of entangled spins are depicted.From the plot it is understandable that the present pre-scription does not hold for N = 1. For N = 2 the shiftsvary rapidly and reach a peak value. Once N increasesthe shift gradually decreases and for large N saturates toa constant value of the normalization at 1 /
2. Howeverthe scaling in these plots is different because of the pres-ence of F ( L, k, ω ) which we have fixed by fixing the L , k and ω . From this plot one can study the N depen-dent behaviour of the shifts. In the first plot of fig. (2),the behaviour of the shifts with respect to the Cosmo-logical Constant are depicted, for given large N . Thereemerge two natural length scales in the problem: onefrom the system, i.e., L which is the Euclidean distancebetween two consecutive neighbouring spins and anotherfrom the bath k , which is related to the curvature andthe cosmological constant. An interplay between these FIG. 2. Behaviour of the spectroscopic shifts with the Cos-mological Constant and euclidean distance. Here we fix µ = 0 . ω = 1 for the large number of entangled spin, N = 50000. - - - - - - - - - - - - - FIG. 3. Behaviour of Cosmological Constant Λ with the Num-ber of entangled spins N at (L, ω , µ ) = (100, 100, 0.001).The x and y axis values read off the exponents to base 10. two scales leads to rich dynamical consequences. For L (cid:28) k ≈ (cid:112) /R = (cid:112) / Λ one can find an inertial framewhere the laws of Minkowski space-time are valid andthe present shifts reduce to the flat space limit result.For L (cid:29) k , the curvature of the static patch of De Sit-ter space-time dominates and plays a non-trivial role inspectral shifts. Here, the spectral shifts vary as L − anddepend explicitly on k . These are related to the Cos-mological Constant Λ and can be further linked to theequilibrium temperature of the bath. For this reason wewill focus on the distances L (cid:29) k to have a non-trivialeffect. For L (cid:28) k , the spectral shifts vary as L − andare independent of k or Λ for which the shifts shouldbe essentially the same, as obtained in Minkowski case.Presence of k in the shifts for L (cid:29) k confirms the pres-ence of Λ in the De Sitter static patch, which is of course,not present in the other limit i.e. L (cid:28) k . We have found,Λ ∼ O (10 − ) in the Planckian unit, this corresponds to almost constant shifts, which is consistent with theobserved value, Λ observed ∼ . × − in Planckianunit [52]. On the other hand, Cosmological Constant inthe region Λ (cid:38) (0 .
05) is not allowed, as it gives an initialoscillation with a very small but fast decaying amplitudeof the shifts. After crossing this region all the shifts ap-proach to zero asymptotically from which we will not getany information of Λ. So the observationally relevant fea-ture will come from the very small Λ where all shifts varyvery slowly in the L (cid:29) k case. Additionally, using thepresent analysis one can further constrain the curvatureof the static patch at very tiny value, R ∼ O (10 − ), cor-responding to Λ ∼ O (10 − ). In the plot of fig. (2), thebehaviour of the shifts with respect to the euclidean dis-tance ( L ) is depicted, for given large N and for the fixedvalue of Cosmological Constant at the observed value. Itis clearly observed from the plot that the shifts for verysmall value of L fluctuates with large amplitude and aswe increase the value of L all of them decay very fastand for the asymptotic large value of L they saturates tonegligibly small value. Further, in fig. (3), we study thebehaviour of Cosmological Constant, Λ, as the numberof entangled spins, N , is varied from very small values of O (1) upto macroscopic values of O (10 ). This kind ofstudy can be used to estimate the thermodynamic limitof number of entangled spins corresponding to the ob-served value of cosmological constant and the curvature.It is also observed that as Λ → − the number of en-tangled spins are of ∼ O (10 ). These highlight the ther-modynamic limit of the system of entangled spins. Thethermodynamic limit lies in the range of O (10 − ).In conclusion, we have studied indirect detection mech-anism of observationally relevant Cosmological Constantfrom the shifts obtained from a realistic model of opensystem consisting of entangled large N spins. For thispurpose, we have utilized the superposition principlealong with equal Euclidean distance between all thespins. In the large N limit-(a) we have found that theshifts are very less sensitive to N , (b) a correct predictionof the observationally consistent Cosmological Constant[52] can be made in the region where the Euclidean dis-tance between all the spins are large enough comparedto the length scale k (i.e. L (cid:29) k ), which implies verytiny value of Cosmological Constant, Λ corresponding tolarge N ∼ O (10 − ) value of the spins and (c) flatspace effects are dominant in the region where the eu-clidean distance between all the spins are small enoughcompared to the length scale k (i.e. L (cid:28) k ). Acknowledgement:
SC would like to thank MaxPlanck Institute for Gravitational Physics, Potsdam forproviding the Post-Doctoral Fellowship. SP acknowl-edges the J. C. Bose National Fellowship for support ofhis research. SC, NG, RND would like to thank NISERBhubaneswar, IISER Mohali and IIT Bombay respec-tively for providing fellowships. Last but not the least,we would like to acknowledge our debt to the people be-longing to the various part of the world for their generousand steady support for research in natural sciences.
Important note:
A detailed supplementary materialis added just after the reference to clarify all the back-ground material related to the present research problem.Few more additional plots and results are also have dis-cussed in this supplementary material to strengthen ourstudy. ∗ Corresponding author,
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A. Computation of N spin Wightman functions FIG. 4. Schwinger Keldysh contour for computing N spinWightman Functions. To compute the N spin Wightman functions of theprobe massless scalar field present in the external ther-mal bath we use the 4D static De Sitter geometry ofour space-time as mentioned earlier. In this coordinatesystem, the equation of motion of the massless externalprobe scalar field can be written as: (cid:34) (cid:0) tα (cid:1) ∂ t (cid:18) cosh (cid:18) tα (cid:19) ∂ t (cid:19) − α cosh (cid:0) tα (cid:1) L (cid:35) Φ( t, χ, θ, φ ) = 0 , (15)where L is the Laplacian operator , which is defined as: L = 1sin χ (cid:20) ∂∂χ (cid:18) sin χ ∂∂χ (cid:19) + 1sin θ ∂∂θ (cid:18) sin θ ∂∂θ (cid:19) + 1sin θ ∂ ∂φ (cid:21) , (16)where χ is related to the radial coordinate r as, r = sin χ. Further, the complete solution for the massless scalarfield is given by:Φ( t, r, θ, φ ) = ∞ (cid:88) l =0 + l (cid:88) m = − l Φ lm ( t, r, θ, φ )= (cid:90) ∞−∞ dω π α √ πω ∞ (cid:88) l =0 + l (cid:88) m = − l Y lm ( θ, φ ) e − iωt (cid:12)(cid:12)(cid:12)(cid:12) Γ ( l + ) Γ( iαω )Γ ( l +3+ iαω ) Γ ( l + iαω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:40) Γ (cid:0) l + (cid:1) Γ( iαω )Γ (cid:0) l +3+ iαω (cid:1) Γ (cid:0) l + iαω (cid:1) (cid:18) r α (cid:19) iαω + Γ ∗ (cid:0) l + (cid:1) Γ ∗ ( iαω )Γ ∗ (cid:0) l +3+ iαω (cid:1) Γ ∗ (cid:0) l + iαω (cid:1) (cid:18) r α (cid:19) − iαω (cid:41) . (17)Next, using this classical solution of the field equationthe quantum field by the following equation:ˆΦ( t, r, θ, φ ) = ∞ (cid:88) l =0 + l (cid:88) m = − l [ a lm Φ lm ( t, r, θ, φ )+ a † lm Φ ∗ lm ( t, r, θ, φ ) (cid:105) . (18)where the quantum states are defined through the fol-lowing condition, a lm | Ψ (cid:105) = 0 , where l = 0 , · · · , ∞ ; m = − l, · · · , + l. (19)Here a lm and a † lm represent the annihilation and creationoperator of the quantum thermal vacuum state | Ψ (cid:105) whichis defined in the bath.Now, we define the consecutive distance between anytwo identical static spins localized at the coordinates( r, θ, φ ) and ( r, θ (cid:48) , φ ) as:∆ z = (cid:88) i =1 ( z i − z (cid:48) i ) = (cid:0) α − r (cid:1) (cid:34) cosh (cid:18) tα (cid:19) − cosh (cid:32) t (cid:48) α (cid:33)(cid:35) + L , (20)Here L represents the euclidean distance between the anytwo identical spins which is defined as, L = 2 r sin (cid:18) ∆ θ (cid:19) , (21)where, ∆ θ is defined as, ∆ θ = θ − θ (cid:48) . Further, the N spin Wightman function for masslessprobe scalar field can be expressed as: G N ( x, x (cid:48) ) = G δδ ( x, x (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) Auto − Correlation G δη ( x, x (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) Cross − Correlation G ηδ ( x, x (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) Cross − Correlation G ηη ( x, x (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) Auto − Correlation β = (cid:104) ˆΦ( x δ , τ )Φ( x δ , τ (cid:48) ) (cid:105) β (cid:104) ˆΦ( x δ , τ )Φ( x η , τ (cid:48) ) (cid:105) β (cid:104) ˆΦ( x η , τ )Φ( x δ , τ (cid:48) ) (cid:105) β (cid:104) ˆΦ( x η , τ )Φ( x η , τ (cid:48) ) (cid:105) β , ∀ δ, η = 1 , · · · , N (for both even & odd) . (22)where the individual Wightman functions can be com-puted using the well known Schwinger Keldysh path in-tegral technique as: G δδ ( x, x (cid:48) ) = G ηη ( x, x (cid:48) )= Tr (cid:104) ρ B ˆΦ( x δ , τ ) ˆΦ( x δ , τ (cid:48) ) (cid:105) = (cid:104) Ψ | ρ B ˆΦ( x δ , τ ) ˆΦ( x δ , τ (cid:48) ) | Ψ (cid:105) = − π { ( z − z (cid:48) ) − ( z − z (cid:48) ) − i(cid:15) } = − π k (cid:0) ∆ τ k − i(cid:15) (cid:1) , (23) G δη ( x, x (cid:48) ) = G ηδ ( x, x (cid:48) )= Tr (cid:104) ρ B ˆΦ( x η , τ ) ˆΦ( x δ , τ (cid:48) ) (cid:105) = (cid:104) Ψ | ρ B ˆΦ( x η , τ ) ˆΦ( x δ , τ (cid:48) ) | Ψ (cid:105) = − π z − z (cid:48) ) − ∆ z − i(cid:15) = − π k (cid:8) sinh (cid:0) ∆ τ k − i(cid:15) (cid:1) − r k sin (cid:0) ∆ θ (cid:1)(cid:9) , (24)where we use the result, sinh (cid:0) ∆ τ k − i(cid:15) (cid:1) ∼ sinh (cid:0) ∆ τ k (cid:1) − i(cid:15) cosh (cid:0) ∆ τ k (cid:1) . Here the thermal density matrix at the bathis defined as: ρ B = exp ( − βH B ) /Z B (25)where H B is the bath Hamiltonian of the massless scalarfield which is defined in Eqn. (3) and Z B is the partitionfunction of the massless scalar field placed at the thermalbath, defined as: Z B = Tr [exp ( − βH B )] = (cid:104) Ψ | exp ( − βH B ) | Ψ (cid:105) . (26)Here | Ψ (cid:105) is the Bunch Davies thermal state of the bathwhich is used to compute the trace operation to deter-mine the individual entries of the Wightman functionsusing Schwinger-Keldysh technique. However, this resultcan be generalised to any non Bunch Davies state (forexample, α vacua). Additionally, we define the followingquantities: k = √ g α = (cid:112) α − r , (27)∆ τ = √ g ( t − t (cid:48) ) = k (cid:18) t − t (cid:48) α (cid:19) , (28) where τ is the proper-time and the length scale k = (cid:112) /R represents the inverse of curvature in De Sitterstatic patch. B. Computation of Hilbert transformation of N spinWightman functions Now, using the Hilbert transformations one can easilyfix the elements of the effective Hamiltonian matrix H ( δη ) ij as appearing in the Lamb Shift part of the Hamiltonian: H ( δη ) ij = H ( ηδ ) ij = (cid:26) D δδ δ ij − i Q δδ (cid:15) ijk δ k − D δδ δ i δ j , δ = η D δη δ ij − i Q δη (cid:15) ijk δ k − D δη δ i δ j . δ (cid:54) = η (29)where we define: D δδ = µ (cid:104) K ( δδ ) ( ω ) + K ( δδ ) ( − ω ) (cid:105) (30) Q δδ = µ (cid:104) K ( δδ ) ( ω ) − K ( δδ ) ( − ω ) (cid:105) , (31) D δη = µ (cid:104) K ( δη ) ( ω ) + K ( δη ) ( − ω ) (cid:105) , (32) Q δη = µ (cid:104) K ( δη ) ( ω ) − K ( δη ) ( − ω ) (cid:105) , (33)where K δη ( ± ω ) ∀ ( δ, η = 1 , · · · , N ) represents the Hilberttransform of the Wightman functions which can be com-puted as: K δδ ( ± ω ) = P π i (cid:90) ∞−∞ dω ω ∓ ω ω − e πkω , (34) K δη ( ± ω ) = P π i (cid:90) ∞−∞ dω T ( ω, L/ ω ∓ ω ω − e πkω . (35)Here, P represents the principal part of the each inte-grals. For simplicity we also define frequency and eu-clidean distance dependent a new function T ( ω, L/
2) as: T ( ω, L/
2) = sin (cid:0) kω sinh − ( L/ k ) (cid:1) Lω (cid:113) L/ k ) . (36)Finally, substituting the these above mentioned expres-sions and using Bethe regularisation technique we get thefollowing simplified results: H ( δη ) ij = H ( ηδ ) ij = µ P π i × (cid:90) ∞−∞ dω ω { ( δ ij − δ i δ j ) ω − i(cid:15) ijk δ k ω } (1 − e − πkω ) ( ω + ω ) ( ω − ω ) = 0 , δ = η (cid:90) ∞−∞ dω ω { ( δ ij − δ i δ j ) ω − i(cid:15) ijk δ k ω } T ( ω, L/ − e − πkω ) ( ω + ω ) ( ω − ω )= 2 πL (cid:113) L/ k ) cos (cid:0) kω sinh − ( L/ k ) (cid:1) = 16 π µ F ( L, k, ω ) . δ (cid:54) = η (37)where the function F ( L, k, ω ) is defined in Eqn. (111). Hence these matrix elements are fixed which will beneeded for the further computation of the spectroscopicshifts from different possible entangled states for the N spin system under consideration. C. Entangled states for N = 2 (even) and N = 3 (odd) spins For N = 2 case the sets of eigenstates ( | g (cid:105) , | e (cid:105) ) and( | g (cid:105) , | e (cid:105) ) are described by the following expressions: For spin 1 : H = ω (cid:0) σ cos α + σ cos β + σ cos γ (cid:1) Ground state ⇒| g (cid:105) = N − (cid:0) cos α − i cos β (cid:1) γ ⇒ Eigenvalue E (2) G = − ω , (38) Excited state ⇒| e (cid:105) = N (cid:0) cos α + i cos β (cid:1) γ ⇒ Eigenvalue E (2) E = ω . (39) For spin 2 : H = ω (cid:0) σ cos α + σ cos β + σ cos γ (cid:1) Ground state ⇒| g (cid:105) = N − (cid:0) cos α − i cos β (cid:1) γ ⇒ Eigenvalue E (2) G = − ω , (40) Excited state ⇒| e (cid:105) = N (cid:0) cos α + i cos β (cid:1) γ ⇒ Eigenvalue E (2) E = ω , (41)where we define the normalisation factor for spin 1 and2 as: N = 1 √ (cid:112) γ , (42) N = 1 √ (cid:112) γ . (43)Consequently, the ground ( | G (cid:105) ), excited ( | E (cid:105) ), symmet-ric ( | S (cid:105) ) and the anti-symmetric ( | A (cid:105) ) state of the two-entangled spin system can be expressed by the following expression: Ground state : ⇒| G (cid:105) = | g (cid:105) ⊗ | g (cid:105) = N , − (cid:0) cos α − i cos β (cid:1) γ (cid:0) cos α − i cos β (cid:1) γ − (cid:0) cos α − i cos β (cid:1) γ − (cid:0) cos α − i cos β (cid:1) γ , (44) Excited state : ⇒| E (cid:105) = | e (cid:105) ⊗ | e (cid:105) = N , (cid:0) cos α + i cos β (cid:1) γ (cid:0) cos α + i cos β (cid:1) γ (cid:0) cos α + i cos β (cid:1) γ (cid:0) cos α + i cos β (cid:1) γ , , (45) Symmetric state : ⇒| S (cid:105) = 1 √ | e (cid:105) ⊗ | g (cid:105) + | g (cid:105) ⊗ | e (cid:105) ]= N , √ − (cid:0) cos α − i cos β (cid:1) γ − (cid:0) cos α − i cos β (cid:1) γ − (cid:0) cos α − i cos β (cid:1) γ (cid:0) cos α + i cos β (cid:1) γ − (cid:0) cos α + i cos β (cid:1) γ (cid:0) cos α − i cos β (cid:1) γ (cid:0) cos α + i cos β (cid:1) γ + (cid:0) cos α + i cos β (cid:1) γ , , (46) Antisymmetric state : ⇒| A (cid:105) = 1 √ | e (cid:105) ⊗ | g (cid:105) − | g (cid:105) ⊗ | e (cid:105) ]= N , √ (cid:0) cos α − i cos β (cid:1) γ − (cid:0) cos α − i cos β (cid:1) γ (cid:0) cos α − i cos β (cid:1) γ (cid:0) cos α + i cos β (cid:1) γ − − (cid:0) cos α + i cos β (cid:1) γ (cid:0) cos α − i cos β (cid:1) γ (cid:0) cos α + i cos β (cid:1) γ − (cid:0) cos α + i cos β (cid:1) γ , , (47)where we define the two spin normalisation factor N , as: N , = N N = 12 (cid:112) (1 + cos γ )(1 + cos γ ) . (48)For N = 3 case for the third spin the sets of eigen-states ( | g (cid:105) , | e (cid:105) ) are described by the following expres-sions: sets of eigenstates ( | g (cid:105) , | e (cid:105) ) and ( | g (cid:105) , | e (cid:105) ) aredescribed by the following expressions: For spin 1 : H = ω (cid:0) σ cos α + σ cos β + σ cos γ (cid:1) Ground state ⇒| g (cid:105) = N − (cid:0) cos α − i cos β (cid:1) γ ⇒ Eigenvalue E (2) G = − ω , (49) Excited state ⇒| e (cid:105) = N (cid:0) cos α + i cos β (cid:1) γ ⇒ Eigenvalue E (2) E = ω . (50) For spin 2 : H = ω (cid:0) σ cos α + σ cos β + σ cos γ (cid:1) Ground state ⇒| g (cid:105) = N − (cid:0) cos α − i cos β (cid:1) γ ⇒ Eigenvalue E (2) G = − ω , (51) Excited state ⇒| e (cid:105) = N (cid:0) cos α + i cos β (cid:1) γ ⇒ Eigenvalue E (2) E = ω , (52) For spin 3 : H = ω (cid:0) σ cos α + σ cos β + σ cos γ (cid:1) Ground state ⇒| g (cid:105) = N − (cid:0) cos α − i cos β (cid:1) γ ⇒ Eigenvalue E (3) G = − ω , (53) Excited state ⇒| e (cid:105) = N (cid:0) cos α + i cos β (cid:1) γ ⇒ Eigenvalue E (3) E = ω . (54)where we define the normalisation factor for spin 1, 2 and3 as: N δ = 1 √ (cid:112) γ . (55)Consequently, the ground ( | G (cid:105) ), excited ( | E (cid:105) ), symmetric( | S (cid:105) ) and the anti-symmetric ( | A (cid:105) ) state of the three-entangled spin system can be expressed as: Ground state : ⇒| G (cid:105) = 1 √ | g (cid:105) ⊗ | g (cid:105) + | g (cid:105) ⊗ | g (cid:105) + | g (cid:105) ⊗ | g (cid:105) ] = 12 √ (cos( α − i cos( β α − i cos( β (cid:112) cos( γ (cid:112) cos( γ α − i cos( β α − i cos( β (cid:112) cos( γ (cid:112) cos( γ α − i cos( β α − i cos( β (cid:112) cos( γ (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ , (56) Excited state : ⇒| E (cid:105) = 1 √ | e (cid:105) ⊗ | e (cid:105) + | e (cid:105) ⊗ | e (cid:105) + | e (cid:105) ⊗ | e (cid:105) ] = 12 √ (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ α i cos( β α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β α i cos( β (cid:112) cos( γ (cid:112) cos( γ , , (57)0 Symmetric state : ⇒| S (cid:105) = 1 √ | e (cid:105) ⊗ | g (cid:105) + | g (cid:105) ⊗ | e (cid:105) + | e (cid:105) ⊗ | g (cid:105) + | g (cid:105) ⊗ | e (cid:105) + | e (cid:105) ⊗ | g (cid:105) + | g (cid:105) ⊗ | e (cid:105) ]= 12 √ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cos( α − i cos( β α i cos( β (cid:112) cos( γ (cid:112) cos( γ − (cos( α − i cos( β α i cos( β (cid:112) cos( γ (cid:112) cos( γ − (cos( α − i cos( β α i cos( β (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ
1) + 1 (cid:112) cos( γ
2) + 1 + (cid:112) cos( γ
1) + 1 (cid:112) cos( γ
3) + 1 + (cid:112) cos( γ
2) + 1 (cid:112) cos( γ
3) + 1 − (cos( α i cos( β α − i cos( β (cid:112) cos( γ (cid:112) cos( γ − (cos( α i cos( β α − i cos( β (cid:112) cos( γ (cid:112) cos( γ − (cos( α i cos( β α − i cos( β (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ
1) + 1 (cid:112) cos( γ
2) + 1 + (cid:112) cos( γ
1) + 1 (cid:112) cos( γ
3) + 1 + (cid:112) cos( γ
2) + 1 (cid:112) cos( γ
3) + 1 (cid:112) cos( γ α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ , , (58) Antisymmetric state : ⇒| A (cid:105) = 1 √ | e (cid:105) ⊗ | g (cid:105) − | g (cid:105) ⊗ | e (cid:105) + | e (cid:105) ⊗ | g (cid:105) − | g (cid:105) ⊗ | e (cid:105) + | e (cid:105) ⊗ | g (cid:105) − | g (cid:105) ⊗ | e (cid:105) ]= 12 √ (cid:112) cos( γ α − i cos( β (cid:112) cos( γ (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ − (cid:112) cos( γ α − i cos( β (cid:112) cos( γ α − i cos( β α i cos( β (cid:112) cos( γ (cid:112) cos( γ α − i cos( β α i cos( β (cid:112) cos( γ (cid:112) cos( γ α − i cos( β α i cos( β (cid:112) cos( γ (cid:112) cos( γ (cid:112) cos( γ
1) + 1 (cid:112) cos( γ
2) + 1 + (cid:112) cos( γ
1) + 1 (cid:112) cos( γ
3) + 1 + (cid:112) cos( γ
2) + 1 (cid:112) cos( γ
3) + 1 − (cos( α i cos( β α − i cos( β (cid:112) cos( γ (cid:112) cos( γ − (cos( α i cos( β α − i cos( β (cid:112) cos( γ (cid:112) cos( γ − (cos( α i cos( β α − i cos( β (cid:112) cos( γ (cid:112) cos( γ − (cid:112) cos( γ
1) + 1 (cid:112) cos( γ
2) + 1 − (cid:112) cos( γ
1) + 1 (cid:112) cos( γ
3) + 1 − (cid:112) cos( γ
2) + 1 (cid:112) cos( γ
3) + 1 (cid:112) cos( γ α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ − (cid:112) cos( γ α i cos( β (cid:112) cos( γ (cid:112) cos( γ α i cos( β (cid:112) cos( γ − (cid:112) cos( γ α i cos( β (cid:112) cos( γ − (cid:112) cos( γ α i cos( β (cid:112) cos( γ , , (59) D. Direction cosine dependent angular distributionfactors for N = 2 (even) and N = 3 (odd) spins For N = 2 case we have two angular distribution Γ DC and Γ DC , which are defined as:Γ DC = Ω (cid:8) ( B + C − A − D ) cos (cid:0) α (cid:1) cos (cid:0) α (cid:1) + ( A + B + C + D ) cos (cid:0) β (cid:1) cos (cid:0) β (cid:1)(cid:9) , (60)Γ DC = Ω (cid:110) ( ˜ D + ˜ A − ˜ B − ˜ C ) cos (cid:0) α (cid:1) cos (cid:0) α (cid:1) − ( ˜ A + ˜ B + ˜ C + ˜ D ) cos (cid:0) β (cid:1) cos (cid:0) β (cid:1)(cid:111) , (61)where we define few quantities important for rest of thecalculation: A = (cid:20) cos α − i cos β γ + cos α − i cos β γ (cid:21) (62) B = (cid:20) − cos α − i cos β γ . cos α + i cos β γ (cid:21) (63) C = (cid:20) − cos α + i cos β γ . cos α − i cos β γ (cid:21) (64) D = (cid:20) cos α + i cos β γ + cos α + i cos β γ (cid:21) (65) ˜ A = (cid:20) cos α − i cos β γ − cos α − i cos β γ (cid:21) , (66)˜ B = (cid:20) α − i cos β γ cos α + i cos β γ (cid:21) , (67)˜ C = (cid:20) − − cos α + i cos β γ cos α − i cos β γ (cid:21) , (68)˜ D = (cid:20) cos α − i cos β γ − cos α + i cos β γ (cid:21) , (69)Ω = 12 √ (cid:112) (1 + cos γ )(1 + cos γ ) = N N = N , . (70)For N = 3 case we introduce few symbols to write theangular dependence of the spectral shiftΩ = 12 (cid:112) γ , (71)Ω = 12 (cid:112) γ , (72)Ω = 12 (cid:112) γ , (73) α = cos α cos α , (74) β = cos β cos β , (75)˜ α = cos α − i cos β , (76)1˜ α = cos α − i cos β , (77)˜ α = cos α − i cos β , (78)˜ α ∗ = cos α + i cos β , (79)˜ α ∗ = cos α + i cos β , (80)˜ α ∗ = cos α + i cos β (81)Therefore the angular dependence for the ground statein this case can be written as:Γ DC = 16 ( G + G + G + G ) (82)where we define: G = 2(Ω Ω + Ω Ω + Ω Ω ) (cid:20) − i ( α − β ) (cid:18) ˜ α ˜ α Ω + ˜ α ˜ α Ω + ˜ α ˜ α Ω (cid:19) + 2 i ( α + iβ ) (cid:18) ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19) + 2 i ( α + iβ ) (cid:18) ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19)(cid:21) , (83) G = (cid:18) ˜ α ∗ ˜ α ∗ Ω + ˜ α ∗ ˜ α ∗ Ω + ˜ α ∗ ˜ α ∗ Ω (cid:19)(cid:20) i ( α − iβ ) (cid:18) ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19) + 2 i ( α − iβ ) (cid:18) ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19) − i ( α − β )(2Ω Ω + 2Ω Ω + 2Ω Ω )] , (84) G = (cid:18) ˜ α ∗ Ω − ˜ α ∗ Ω − ˜ α ∗ Ω (cid:19)(cid:20) − i ( α + iβ ) (cid:18) ˜ α ˜ α Ω + ˜ α ˜ α Ω + ˜ α ˜ α Ω (cid:19) + 2 i ( α + β ) (cid:18) ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19) − i ( α − iβ ) (2Ω Ω + 2Ω Ω + 2Ω Ω )] , (85) G = − (cid:18) ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω (cid:19)(cid:20) − i ( α + iβ ) (cid:18) ˜ α ˜ α Ω + ˜ α ˜ α Ω + ˜ α ˜ α Ω (cid:19) − i ( α + β ) (cid:18) ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19) − i ( α − iβ ) (2Ω Ω + 2Ω Ω + 2Ω Ω )] . (86)Therefore the angular dependence for the excited statein this case can be written as:Γ DC = 16 ( E + E + E + E ) (87) where we define: E = 2(Ω Ω + Ω Ω + Ω Ω ) (cid:20) − i ( α − β ) (cid:18) ˜ α ∗ ˜ α ∗ Ω + ˜ α ∗ ˜ α ∗ Ω + ˜ α ∗ ˜ α ∗ Ω (cid:19) − i ( α − iβ ) (cid:18) ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω (cid:19) − i ( α − iβ ) (cid:18) ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω (cid:19)(cid:21) , (88) E = (cid:18) ˜ α ˜ α Ω + ˜ α ˜ α Ω + ˜ α ˜ α Ω (cid:19)(cid:20) − i ( α + iβ ) (cid:18) ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω (cid:19) − i ( α + iβ ) (cid:18) ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω (cid:19) − i ( α − β )(2Ω Ω + 2Ω Ω + 2Ω Ω )] , (89) E = (cid:18) ˜ α Ω − ˜ α Ω − ˜ α Ω (cid:19)(cid:20) − i ( α − iβ ) (cid:18) ˜ α ∗ ˜ α ∗ Ω + ˜ α ∗ ˜ α ∗ Ω + ˜ α ∗ ˜ α ∗ Ω (cid:19) − i ( α + β ) (cid:18) ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω (cid:19) − i ( α + iβ ) (2Ω Ω + 2Ω Ω + 2Ω Ω )] , (90) E = (cid:18) ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19)(cid:20) − i ( α − iβ ) (cid:18) ˜ α ∗ ˜ α ∗ Ω + ˜ α ∗ ˜ α ∗ Ω + ˜ α ∗ ˜ α ∗ Ω (cid:19) − i ( α + β ) (cid:18) ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω (cid:19) − i ( α + iβ ) (2Ω Ω + 2Ω Ω + 2Ω Ω )] , (91)Therefore the angular dependence for the Symmetricstate in this case can be written as:Γ DC = 16 ( S + S + S + S ) (92)where we define: S = (cid:18) − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω + 2Ω Ω + 2Ω Ω + 2Ω Ω (cid:19)(cid:20) − i ( α + iβ ) (cid:18) − ˜ α Ω − ˜ α Ω − ˜ α Ω − ˜ α Ω − ˜ α Ω − ˜ α Ω (cid:19) − i ( α − iβ ) (cid:18) ˜ α Ω + ˜ α Ω + ˜ α Ω + ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19) − i ( α + β ) (cid:18) − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω + 2Ω Ω + 2Ω Ω + 2Ω Ω )] , (93)2 S = (cid:18) − ˜ α ∗ Ω − ˜ α ∗ Ω − ˜ α ∗ Ω − ˜ α ∗ Ω − ˜ α ∗ Ω − ˜ α ∗ Ω (cid:19)(cid:20) − i ( α − β ) (cid:18) ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω (cid:19) − i ( α − iβ ) (cid:18) − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω + 2Ω Ω + 2Ω Ω + 2Ω Ω ) − i ( α − iβ ) (cid:18) − ˜ α ˜ α ∗ Ω − ˜ α ˜ α ∗ Ω − ˜ α ˜ α ∗ Ω + 2Ω Ω + 2Ω Ω + 2Ω Ω )] (94) S = (cid:18) ˜ α Ω + ˜ α Ω + ˜ α Ω + ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19)(cid:20) − i ( α − β ) (cid:18) − ˜ α Ω − ˜ α Ω − ˜ α Ω − ˜ α Ω − ˜ α Ω − ˜ α Ω (cid:19) − i ( α + iβ ) (cid:18) − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω + 2Ω Ω + 2Ω Ω + 2Ω Ω ) − i ( α + iβ ) (cid:18) − ˜ α ˜ α ∗ Ω − ˜ α ˜ α ∗ Ω − ˜ α ˜ α ∗ Ω + 2Ω Ω + 2Ω Ω + 2Ω Ω )] (95) S = (cid:18) − ˜ α ˜ α ∗ Ω − ˜ α ˜ α ∗ Ω − ˜ α ˜ α ∗ Ω + 2Ω Ω + 2Ω Ω + 2Ω Ω ) (cid:20) − i ( α + iβ ) (cid:18) − ˜ α Ω − ˜ α Ω − ˜ α Ω − ˜ α Ω − ˜ α Ω − ˜ α Ω (cid:19) − i ( α − iβ ) (cid:18) ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω (cid:19) − i ( α + β ) (cid:18) − ˜ α ˜ α ∗ Ω − ˜ α ˜ α ∗ Ω − ˜ α ˜ α ∗ Ω + 2Ω Ω + 2Ω Ω + 2Ω Ω )] (96)Therefore,the angular dependence for the Antisymmetricstate in this case can be written as:Γ DC = 16 ( A + A + A + A ) (97) where we define: A = (cid:18) ˜ α ∗ ˜ α Ω + ˜ α ∗ ˜ α Ω + ˜ α ∗ ˜ α Ω + 2Ω Ω + 2Ω Ω + 2Ω Ω (cid:19)(cid:20) − i ( α + iβ ) (cid:18) − ˜ α Ω + ˜ α Ω − ˜ α Ω − ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19) − i ( α − iβ ) (cid:18) − ˜ α Ω + ˜ α Ω + ˜ α Ω − ˜ α Ω − ˜ α Ω + ˜ α Ω (cid:19) − i ( α + β ) (cid:18) − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω − Ω − Ω − Ω )] , (98) A = (cid:18) − ˜ α ∗ Ω + ˜ α ∗ Ω − ˜ α ∗ Ω − ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω (cid:19)(cid:20) − i ( α − β ) (cid:18) − ˜ α ∗ Ω + ˜ α ∗ Ω − ˜ α ∗ Ω − ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω (cid:19) − i ( α − iβ ) (cid:18) − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω − Ω − Ω − Ω ) − i ( α − iβ ) (cid:18) ˜ α ˜ α ∗ Ω + ˜ α ˜ α ∗ Ω + ˜ α ˜ α ∗ Ω + 2Ω Ω + 2Ω Ω + 2Ω Ω )] , (99) A = (cid:18) − ˜ α Ω + ˜ α Ω − ˜ α Ω − ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19)(cid:20) − i ( α − β ) (cid:18) − ˜ α Ω + ˜ α Ω − ˜ α Ω − ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19) − i ( α + iβ ) (cid:18) − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω − ˜ α ∗ ˜ α Ω − Ω − Ω − Ω ) − i ( α + iβ ) (cid:18) ˜ α ˜ α ∗ Ω + ˜ α ˜ α ∗ Ω + ˜ α ˜ α ∗ Ω + 2Ω Ω + 2Ω Ω + 2Ω Ω )] , (100) A = (cid:18) − ˜ α ˜ α ∗ Ω − ˜ α ˜ α ∗ Ω − ˜ α ˜ α ∗ Ω − Ω − Ω − Ω (cid:19)(cid:20) − i ( α + iβ ) (cid:18) − ˜ α Ω + ˜ α Ω − ˜ α Ω − ˜ α Ω + ˜ α Ω + ˜ α Ω (cid:19) − i ( α − iβ ) (cid:18) − ˜ α ∗ Ω + ˜ α ∗ Ω − ˜ α ∗ Ω − ˜ α ∗ Ω + ˜ α ∗ Ω + ˜ α ∗ Ω (cid:19) − i ( α + β ) (cid:18) ˜ α ˜ α ∗ Ω + ˜ α ˜ α ∗ Ω + ˜ α ˜ α ∗ Ω + 2Ω Ω + 2Ω Ω + 2Ω Ω )] . (101)3 D. Spectroscopic shifts for N spins in static patch ofDe Sitter space To compute the spectroscopic shifts from the entangledground, excited, symmetric and antisymmetric states we need to compute the following expressions for N spinsystem: Ground state : δE NG = (cid:104) G | H LS | G (cid:105) = − i N (cid:88) δ,η =1 3 (cid:88) i,j =1 H ( δη ) ij (cid:104) G | ( n δi .σ δi )( n ηj .σ ηj ) | G (cid:105) = − P F ( L, k, ω )Γ N DC N , (102) Excited state : δE NE = (cid:104) E | H LS | E (cid:105) = − i N (cid:88) δ,η =1 3 (cid:88) i,j =1 H ( δη ) ij (cid:104) E | ( n δi .σ δi )( n ηj .σ ηj ) | E (cid:105) = − P F ( L, k, ω )Γ N DC N , (103) Symmetric state : δE NS = (cid:104) S | H LS | S (cid:105) = − i N (cid:88) δ,η =1 3 (cid:88) i,j =1 H ( δη ) ij (cid:104) S | ( n δi .σ δi )( n ηj .σ ηj ) | S (cid:105) = − P F ( L, k, ω )Γ N DC N , , (104) Antisymmetric state : δE NA = (cid:104) A | H LS | A (cid:105) = − i N (cid:88) δ,η =1 3 (cid:88) i,j =1 H ( δη ) ij (cid:104) A | ( n δi .σ δi )( n ηj .σ ηj ) | A (cid:105) = P F ( L, k, ω )Γ N DC N . (105)Here the overall normalisation factor is appearing fromthe N entangled spin states, which is given by, N norm = 1 / (cid:112) N C = (cid:112) N − /N !. For the computation of thematrix elements in the above mentioned shifts we haveused the following results: N (cid:88) δ,η =1 3 (cid:88) i,j =1 (cid:104) G | ( n δi .σ δi )( n ηj .σ ηj ) | G (cid:105) = 1 N N (cid:88) δ,η =1 N (cid:88) δ (cid:48) ,η (cid:48) =1 ,δ (cid:48) <η (cid:48) N (cid:88) δ (cid:48)(cid:48) ,η (cid:48)(cid:48) =1 ,δ (cid:48)(cid:48) <η (cid:48)(cid:48) (cid:88) i,j =1 (cid:104) g η (cid:48) | ⊗ (cid:104) g δ (cid:48) | ( n δi .σ δi )( n ηj .σ ηj ) | g δ (cid:48)(cid:48) (cid:105) ⊗ | g η (cid:48)(cid:48) (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) ≡ Γ N DC = 1 N Γ N DC , (106) N (cid:88) δ,η =1 3 (cid:88) i,j =1 (cid:104) E | ( n δi .σ δi )( n ηj .σ ηj ) | E (cid:105) = 1 N N (cid:88) δ,η =1 N (cid:88) δ (cid:48) ,η (cid:48) =1 ,δ (cid:48) <η (cid:48) N (cid:88) δ (cid:48)(cid:48) ,η (cid:48)(cid:48) =1 ,δ (cid:48)(cid:48) <η (cid:48)(cid:48) (cid:88) i,j =1 (cid:104) e η (cid:48) | ⊗ (cid:104) e δ (cid:48) | ( n δi .σ δi )( n ηj .σ ηj ) | e δ (cid:48)(cid:48) (cid:105) ⊗ | e η (cid:48)(cid:48) (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) ≡ Γ N DC = 1 N Γ N DC , (107) N (cid:88) δ,η =1 3 (cid:88) i,j =1 (cid:104) S | ( n δi .σ δi )( n ηj .σ ηj ) | S (cid:105) = 12 N N (cid:88) δ,η =1 N (cid:88) δ (cid:48) ,η (cid:48) =1 ,δ (cid:48) <η (cid:48) N (cid:88) δ (cid:48)(cid:48) ,η (cid:48)(cid:48) =1 ,δ (cid:48)(cid:48) <η (cid:48)(cid:48) (cid:88) i,j =1 ( (cid:104) e η (cid:48) (cid:105) ⊗ (cid:104) g δ (cid:48) | + (cid:104) g η (cid:48) | ⊗ (cid:104) e δ (cid:48) | ) | ( n δi .σ δi )( n ηj .σ ηj ) | ( | e δ (cid:48)(cid:48) (cid:105) ⊗ | g η (cid:48)(cid:48) (cid:105) + | g δ (cid:48)(cid:48) (cid:105) ⊗ | e η (cid:48)(cid:48) (cid:105) ) (cid:124) (cid:123)(cid:122) (cid:125) ≡ Γ N DC = − N Γ N DC , (108)4 N (cid:88) δ,η =1 3 (cid:88) i,j =1 (cid:104) A | ( n δi .σ δi )( n ηj .σ ηj ) | A (cid:105) = 12 N N (cid:88) δ,η =1 N (cid:88) δ (cid:48) ,η (cid:48) =1 ,δ (cid:48) <η (cid:48) N (cid:88) δ (cid:48)(cid:48) ,η (cid:48)(cid:48) =1 ,δ (cid:48)(cid:48) <η (cid:48)(cid:48) (cid:88) i,j =1 ( (cid:104) e η (cid:48) (cid:105) ⊗ (cid:104) g δ (cid:48) | − (cid:104) g η (cid:48) | ⊗ (cid:104) e δ (cid:48) | ) | ( n δi .σ δi )( n ηj .σ ηj ) | ( | e δ (cid:48)(cid:48) (cid:105) ⊗ | g η (cid:48)(cid:48) (cid:105) − | g δ (cid:48)(cid:48) (cid:105) ⊗ | e η (cid:48)(cid:48) (cid:105) ) (cid:124) (cid:123)(cid:122) (cid:125) ≡ Γ N DC = − N Γ N DC . (109)Here we found from our computation that the directioncosine dependent factors which are coming as an out-come of the · · · (cid:124)(cid:123)(cid:122)(cid:125) highlighted contributions are exactlysame for ground and excited states, so that the shifts arealso appearing to be exactly same with same signature.On the other hand, from the symmetric and antisymmet-ric states we have found that he direction cosine depen-dent highlighted factors are not same. Consequently, theshifts are not also same for these two states. Now onecan fix the principal value of the Hilbert transformedintegral of the N spin Wightman functions to be unity( P = 1) for the sake of simplicity, as it just serves the pur-pose of a overall constant scaling of the computed shiftsfrom all the entangled states for N spins. The explicitexpressions for these direction cosine dependent factorsare extremely complicated to write for any general largevalue of the number of N spins. For this reason we havenot presented these expressions explicitly in this paper.However, for N = 2 and N = 3 spin systems we havepresented the results just in the previous section of thissupplementary material of this paper. Finally, one canwrite the following expression for the ratio of the spec-troscopic shifts with the corresponding direction cosinedependent factor in a compact notation is derived as: δE NY N DC = δE NS Γ N DC = − δE NA Γ N DC = −F ( L, ω , k ) / N , (110)where Y represents the ground and the excited statesand S and A symmetric and antisymmetric states, re-spectively. Here, Γ Ni ; DC ∀ i = 1 , , N number of identical spins. This result explicitlyshows that the ratio of all these shifts with their corre-sponding direction cosine dependent factor proportionalto a spectral function F ( L, ω , k ), given by, F ( L, k, ω ) = E ( L, k ) cos (cid:0) ω k sinh − ( L/ k ) (cid:1) , (111)where, E ( L, k ) = µ / (8 πL (cid:112) L/ k ) ) . Here this spectral function is very important as it is theonly contribution in this computation which actually di-rectly captures the contribution of the static patch of the De Sitter space-time through the parameter k . Inthis computation we are dealing with two crucial lengthscale which are both appearing in the spectral function F ( L, k, ω ), which are:1. Euclidean distance L and2. Parameter k which plays the role of inverse curva-ture in this problem.Depending on these two length scales to analyse the be-haviour of this spectral function we have considered twolimiting situations, which are given by: • Region L (cid:29) k , which is very useful for our compu-tation as it captures the effect of both the lengthscale L and k . We have found that to determinethe observed value of the Cosmological Constant atthe present day in Planckian unit this region givesvery important contribution. • Region L (cid:28) k , which replicates the analogous ef-fect of Minkowski flat space-time in the computa-tion of spectral shifts. This limiting result maynot be very useful for our computation, but clearlyshows that exactly when we will loose all the infor-mation of the static patch of the De Sitter space.For this reason this region is also not useful at all todetermine the value of the observationally consis-tent value of Cosmological Constant from the spec-tral shifts. In the later section of this supplemen-tary material it will be shown that if we start doingthe same computation of spectral shifts in exactlyMinkowski flat space-time then we will get the sameresults of the spectral shifts that we have obtainedin this limiting region.In different euclidean length scales, we have the follow-ing approximated expressions for the above mentionedfunction: F ( L, k, ω ) = µ k πL cos (2 ω k ln ( L/ k )) , L >> kµ πL cos ( ω L ) . L << k (112) E. Large N limit of spectroscopic shifts In this section our objective is to derive the expressionfor shifts at large N limit. This large N limit is very use-ful to describe a realistic system in nature and usuallyidentified to be the thermodynamic limit. Stirling’s ap-proximation is very useful to deal with factorials of verylarge number. The prime reason of using Stirling’s ap-proximation is to estimate a correct numerical value ofthe factorial of very large number, provided small errorwill appear in this computation. However, this is really useful as numerically dealing with the factorial of verylarge number is extremely complicated job to performand in some cases completely impossible to perform. Inour computation this large number is explicitly appear-ing in the normalization constant of the entangled states, N norm = 1 / (cid:112) N C = (cid:112) N − /N !, which we will fur-ther analytically estimate using Stirling’s formula. Now,according to this approximation one can write the expres-sion for the factorial of a very large number (in our con-text that number N correspond to the number of spins)as: Stirling (cid:48) s formula : N ! ∼ √ N π (cid:18) Ne (cid:19) N N + O (cid:18) N (cid:19) + · · · (cid:124) (cid:123)(cid:122) (cid:125) small corrections , (113)which finally leads to the following bound on N !, where N is a positive integer for our system, as: √ π N N + exp( − N ) exp (cid:18) N + 1 (cid:19) ≤ N ! ≤ exp(1) N N + exp( − N ) exp (cid:18) N (cid:19) . (114)Later Gosper had introduced further modification in theStirling’s formula to get more accurate answer of the fac- torial of a very large number, which is given by the fol-lowing expression: Stirling Gosper formula : N ! ∼ (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) N + 13 (cid:124)(cid:123)(cid:122)(cid:125) Gosper factor π (cid:18) Ne (cid:19) N N + O (cid:18) N (cid:19) + · · · (cid:124) (cid:123)(cid:122) (cid:125) small corrections . (115)Using this formula one can further evaluate the expres- sion for ( N − N spin system as:( N − ∼ (cid:115)(cid:18) N − (cid:19) π (cid:18) N − e (cid:19) N − N −
2) + O (cid:18) N − (cid:19) + · · · (cid:124) (cid:123)(cid:122) (cid:125) small corrections . (116)Here we want to point out few more revised version of the Stirling’s formula, which are commonly used in variouscontexts:6 Stirling Burnside formula : N ! ∼ √ π (cid:18) N + e (cid:19) N + , (117) Stirling Ramanujan formula : N ! ∼ √ π (cid:18) Ne (cid:19) N (cid:18) N + 12 N + 18 N + 1240 (cid:19) / , (118) Stirling Windschitl formula : N ! ∼ √ πN (cid:18) Ne (cid:19) N (cid:18) N sinh 1 N (cid:19) N/ , (119) Stirling Nemes formula : N ! ∼ √ πN (cid:18) Ne (cid:19) N (cid:18) N − (cid:19) N . (120)Further in the large N limit, using the Stirling-Gosper approximation, the normalization factor can be writtenas: N norm = 1 (cid:112) N C Large N −−−−−→ (cid:92) N norm ≈ √ (cid:32) − (cid:0) N + (cid:1) (cid:33) / (cid:18) Ne (cid:19) − N/ (cid:18) N − e (cid:19) N/ − (cid:118)(cid:117)(cid:117)(cid:116) − ( N + ) (cid:0) − N (cid:1) . (121)Thus in the large N limit the spectral shifts can be ap-proximately derived as : (cid:91) δE NY N DC = (cid:91) δE NS Γ N DC = − (cid:91) δE NA Γ N DC = −F ( L, k, ω ) / (cid:92) N norm2 . (122)From the above expressions derived in the large N limitwe get the following information: • Contribution from the large N limit will only effectthe normalization factors appearing in the shifts, • The prime contribution, which is comping from thespectral function F ( L, ω , k ) is independent of thenumber N . So it is expected that directly this con- tribution will not be effected by the large N limitingapproximation in the factorial. F. Flat space limit of spectroscopic shifts for N spins Now, our objective is to the obtained results for spec-troscopic shifts in the
L << k limit with the result onecan derive in the context of the Minkowski flat space.Considering the same physical set up, the two point ther-mal correlation functions can be expressed in terms of the N spin Wightman function for massless probe scalar fieldcan be expressed as: G Min N ( x, x (cid:48) ) = G δδ Min ( x, x (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) Auto − Correlation G δη Min ( x, x (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) Cross − Correlation G ηδ Min ( x, x (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) Cross − Correlation G ηη Min ( x, x (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) Auto − Correlation β = (cid:104) ˆΦ( x δ , τ )Φ( x δ , τ (cid:48) ) (cid:105) β (cid:104) ˆΦ( x δ , τ )Φ( x η , τ (cid:48) ) (cid:105) β (cid:104) ˆΦ( x η , τ )Φ( x δ , τ (cid:48) ) (cid:105) β (cid:104) ˆΦ( x η , τ )Φ( x η , τ (cid:48) ) (cid:105) β Min , ∀ δ, η = 1 , · · · , N (for both even & odd) . (123)where the individual Wightman functions can be com- puted using the well known Schwinger Keldysh path in-tegral technique as:7 G δδ Min ( x, x (cid:48) ) = − π ∞ (cid:88) m = −∞ τ − i { πkm + (cid:15) } ) = 116 π k cosec (cid:18) (cid:15) + i ∆ τ k (cid:19) , (124) G δη Min ( x, x (cid:48) ) = − π ∞ (cid:88) m = −∞ τ − i { πkm + (cid:15) } ) − L = 116 π kL (cid:20) (cid:26) Floor (cid:18) π arg (cid:18) (cid:15) + i (∆ τ + L ) k (cid:19)(cid:19) − Floor (cid:18) π arg (cid:18) (cid:15) + i (∆ τ − L ) k (cid:19)(cid:19)(cid:27) + i (cid:26) cot (cid:18) (cid:15) + i (∆ τ + L )2 k (cid:19) − cot (cid:18) (cid:15) + i (∆ τ − L )2 k (cid:19)(cid:27)(cid:21) , (125)where (cid:15) is an infinitesimal quantity which is introducedto deform the contour of the path integration. Using this Wightman function we can carry forward the similar cal-culation for spectroscopic shifts in Minkowsi space, whichgives: For general N : δE NY,
Min N DC = δE NS,
Min Γ N DC = − δE NA,
Min Γ N DC (cid:124) (cid:123)(cid:122) (cid:125) Minkowski space calculation = − cos ( ω L ) / N = δE NY,
Min N DC = δE NS,
Min Γ N DC = − δE NA,
Min Γ N DC (cid:124) (cid:123)(cid:122) (cid:125) Region L (cid:28) k calculation , (126) For large N : (cid:92) δE NY,
Min N DC = (cid:92) δE NS,
Min Γ N DC = − (cid:92) δE NA,
Min Γ N DC (cid:124) (cid:123)(cid:122) (cid:125) Minkowski space calculation = − cos ( ω L ) / (cid:92) N norm2 = (cid:92) δE NY,
Min N DC = (cid:92) δE NS,
Min Γ N DC = − (cid:92) δE NA,
Min Γ N DC (cid:124) (cid:123)(cid:122) (cid:125) Region L (cid:28) k calculation , (127)where Y represents the ground and the excited statesand S and A symmetric and antisymmetric states, re-spectively. Here, Γ Ni ; DC ∀ i = 1 , , N number of identical spins. Here all thequantities in (cid:98) are evaluated at the large N limit by using Stirling Gosper formula as mentioned earlier. Here it isclearly observed that the shifts are independent of thetemperature of the thermal bath, T = 1 / πk and onlydepends on direction cosines and the euclidean distance L . Also we found that this result exactly matches withthe result obtained for the limiting case L (cid:28) kk