Quantum Damped Harmonic Oscillator

Transcription

1 Provisional chapter Chapter 7 Quantum Damped Harmonic Oscillator Quantum Damped Harmonic Oscillator Kazuyuki Fujii Kazuyuki Fujii Additional information is available at the end of the chapter Additional information is available at the end of the chapter 1. Introduction In this chapter we introduce a toy model of Quantum Mechanics with Dissipation. Quantum Mechanics with Dissipation plays a crucial role to understand real world. However, it is not easy to master the theory for undergraduates. The target of this chapter is eager undergraduates in the world. Therefore, a good toy model to understand it deeply is required. The quantum damped harmonic oscillator is just such a one because undergraduates must use master) many fundamental techniques in Quantum Mechanics and Mathematics. That is, harmonic oscillator, density operator, Lindblad form, coherent state, squeezed state, tensor product, Lie algebra, representation theory, Baker Campbell Hausdorff formula, etc. They are jewels" in Quantum Mechanics and Mathematics. If undergraduates master this model, they will get a powerful weapon for Quantum Physics. I expect some of them will attack many hard problems of Quantum Mechanics with Dissipation. The contents of this chapter are based on our two papers [3] and [6]. I will give a clear and fruitful explanation to them as much as I can.. Some preliminaries In this section let us make some reviews from Physics and Mathematics within our necessity..1. From physics First we review the solution of classical damped harmonic oscillator, which is important to understand the text. For this topic see any textbook of Mathematical Physics. The differential equation is given by ẍ+ω x = γẋ ẍ+γẋ+ω x = 0 γ > 0).1) 01 Fujii, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution Licensehttp://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, 013 Fujii; and licensee reproduction InTech. inthis anyis medium, an open provided access article the original distributed workunder is properly the terms cited. of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 134 Advances in Quantum Mechanics Quantum Mechanics where x = xt), ẋ = dx/dt and the mass m is set to 1 for simplicity. In the following we treat only the case ω > γ/ the case ω = γ/ may be interesting). Solutions with complex form) are well known to be x ± t) = e γ ±i ω γ ) )t, so the general solution is given by xt) = αx + t)+ᾱx t) = αe γ +i ) ω γ ) t γ i + ᾱe ω γ )t ) = αe γ +iω 1 γ ω )t ) γ iω + ᾱe 1 γ ω )t ).) where α is a complex number. If γ/ω is small enough we have an approximate solution xt) αe γ +iω)t + ᾱe γ iω)t..3) Next, we consider the quantum harmonic oscillator. This is well known in textbooks of Quantum Mechanics. As standard textbooks of Quantum Mechanics see [] and [11] [] is particularly interesting). For the Hamiltonian H = Hq, p) = 1 p + ω q ).4) where q = qt), p = pt), the canonical equation of motion reads q H p = p, ṗ H q = ω q. From these we recover the equation See.1) with q = x and λ = 0. q = ω q q+ω q = 0. Next, we introduce the Poisson bracket. For A = Aq, p), B = Bq, p) it is defined as {A, B} c A B q p A B p q.5) where {,} c means classical. Then it is easy to see {q, q} c = 0, {p, p} c = 0, {q, p} c = 1..6)

3 Quantum Damped Harmonic Oscillator 135 Quantum Damped Harmonic Oscillator 3 Now, we are in a position to give a quantization condition due to Dirac. Before that we prepare some notation from algebra. Square matrices A and B don t commute in general, so we need the commutator [A, B] = AB BA. Then Dirac gives an abstract correspondence q ˆq, p ˆp which satisfies the condition [ ˆq, ˆq] = 0, [ ˆp, ˆp] = 0, [ ˆq, ˆp] = i h1.7) corresponding to.6). Here h is the Plank constant, and ˆq and ˆp are both Hermite operators on some Fock space a kind of Hilbert space) given in the latter and 1 is the identity on it. Therefore, our quantum Hamiltonian should be H = H ˆq, ˆp) = 1 ˆp + ω ˆq ).8) from.4). Note that a notation H instead of Ĥ is used for simplicity. From now on we consider a complex version. From.4) and.8) we rewrite like Hq, p) = 1 p + ω q ) = ω q + 1 ω p ) = ω q i ω p)q+ i ω p) and H ˆq, ˆp) = ω ˆq + 1 ω ˆp ) = ω = ω { ˆq i ω ˆp) ˆq+ i ω ˆp)+ h ω { ˆq iω ˆp) ˆq+ iω ˆp) iω [ ˆq, ˆp] } } { ω = ω h h ˆq i ω ˆp) ˆq+ i ω ˆp)+ 1 } by use of.7), and if we set we have easily a = [a, a ] = ω h [ ˆq+ i ω ˆp, ˆq i ω ˆp] = ω h by use of.7). As a result we obtain a well known form ω h ˆq i ω ω ˆp), a = h ˆq+ i ˆp).9) ω { i } [ ˆq, ˆp] = ω { i } ω h ω i h = 1 H = ω ha a+ 1 ), [a, a ] = 1..10) Here we used an abbreviation 1/ in place of 1/)1 for simplicity.

4 136 Advances in Quantum Mechanics 4 Quantum Mechanics If we define an operator N = a a which is called the number operator) then it is easy to see the relations [N, a ] = a, [N, a] = a, [a, a ] = 1..11) For the proof a well known formula [AB, C] = [A, C]B+ A[B, C] is used. Note that aa = a a+[a, a ] = N+ 1. The set {a, a, N} is just a generator of Heisenberg algebra and we can construct a Fock space based on this. Let us note that a, a and N are called the annihilation operator, creation one and number one respectively. First of all let us define a vacuum 0. This is defined by the equation a 0 = 0. Based on this vacuum we construct the n state n like n = a ) n n! 0 0 n). Then we can easily prove a n = n+1 n+1, a n = n n 1, N n = n n.1) and moreover can prove both the orthogonality condition and the resolution of unity m n = δ mn, n n = 1..13) For the proof one can use for example a a ) = aaa )a = an+ 1)a = N+ )aa = N+ )N+ 1) by.11), therefore we have 0 a a ) 0 = 0 N+ )N+ 1) 0 =! = = 1. The proof of the resolution of unity may be not easy for readers we omit it here). As a result we can define a Fock space generated by the generator {a, a, N} F = { c n n C } c n <..14) This is just a kind of Hilbert space. On this space the operators = infinite dimensional matrices) a, a and N are represented as

5 Quantum Damped Harmonic Oscillator 137 Quantum Damped Harmonic Oscillator a = N = a a = , a = , ) by use of.1). Note We can add a phase to {a, a } like b = e iθ a, b = e iθ a, N = b b = a a where θ is constant. Then we have another Heisenberg algebra [N, b ] = b, [N, b] = b, [b, b ] = 1. Next, we introduce a coherent state which plays a central role in Quantum Optics or Quantum Computation. For z C the coherent state z F is defined by the equation a z = z z and z z = 1. The annihilation operator a is not hermitian, so this equation is never trivial. For this state the following three equations are equivalent : 1) a z = z z and z z = 1, ) z = e za za 0, 3) z = e z zn n. n!.16) The proof is as follows. From 1) to ) we use a popular formula e A Be A = B+[A, B]+ 1 [A,[A, B]]+!

6 138 Advances in Quantum Mechanics 6 Quantum Mechanics A, B : operators) to prove e za za) ae za za = a+z. From ) to 3) we use the Baker-Campbell-Hausdorff formula see for example [17]) e A e B = e A+B+ 1 [A,B]+ 1 6 [A,[A,B]]+ 1 6 [B,[A,B]]+. If [A,[A, B]] = 0 = [B,[A, B]] namely, [A, B] commutes with both A and B) then we have e A e B = e A+B+ 1 [A,B] = e 1 [A,B] e A+B = e A+B = e 1 [A,B] e A e B..17) In our case the condition is satisfied because of [a, a ] = 1. Therefore we obtain a famous) decomposition The remaining part of the proof is left to readers. e za za = e z e za e za..18) From the equation 3) in.16) we obtain the resolution of unity for coherent states dxdy π z z = n n = 1 z = x+iy)..19) The proof is reduced to the following formula dxdy π e z z m z n = n! δ mn z = x+iy). The proof is left to readers. See [14] for more general knowledge of coherent states... From mathematics We consider a simple matrix equation where d X = AXB.0) dt X = Xt) = ) x11 t) x 1 t), A = x 1 t) x t) a11 a 1 a 1 a ), B = b11 b 1 b 1 b ).

7 Quantum Damped Harmonic Oscillator 139 Quantum Damped Harmonic Oscillator 7 A standard form of linear differential equation which we usually treat is d dt x = Cx where x = xt) is a vector and C is a matrix associated to the vector. Therefore, we want to rewrite.0) into a standard form. For the purpose we introduce the Kronecker product of matrices. For example, it is defined as ) ) a11 a A B = 1 a11 B a B 1 B a 1 a a 1 B a B a 11 b 11 a 11 b 1 a 1 b 11 a 1 b 1 = a 11 b 1 a 11 b a 1 b 1 a 1 b a 1 b 11 a 1 b 1 a b 11 a b 1.1) a 1 b 1 a 1 b a b 1 a b for A and B above. Note that recently we use the tensor product instead of the Kronecker product, so we use it in the following. Here, let us list some useful properties of the tensor product 1) A 1 B 1 )A B ) = A 1 A B 1 B, ) A E)E B) = A B = E B)A E), 3) e A E+E B = e A E e E B = e A E)E e B ) = e A e B,.) 4) A B) = A B where E is the unit matrix. The proof is left to readers. [9] is recommended as a general introduction. Then the equation.0) can be written in terms of components as or in a matrix form d dt dx 11 dt = a 11 b 11 x 11 + a 11 b 1 x 1 + a 1 b 11 x 1 + a 1 b 1 x, dx 1 dt = a 11 b 1 x 11 + a 11 b x 1 + a 1 b 1 x 1 + a 1 b x, dx 1 dt = a 1 b 11 x 11 + a 1 b 1 x 1 + a b 11 x 1 + a b 1 x, dx dt = a 1 b 1 x 11 + a 1 b x 1 + a b 1 x 1 + a b x x 11 a 11 b 11 a 11 b 1 a 1 b 11 a 1 b 1 x 11 x 1 x 1 = a 11 b 1 a 11 b a 1 b 1 a 1 b x 1 a 1 b 11 a 1 b 1 a b 11 a b 1. x a 1 b 1 a 1 b a b 1 a b x 1 x

8 140 Advances in Quantum Mechanics 8 Quantum Mechanics If we set ) x11 x X = 1 x 1 x = X = x 11, x 1, x 1, x ) T where T is the transpose, then we obtain a standard form from.1). Similarly we have a standard form d dt X = AXB = d dt X = A B T ) X.3) where E T = E for the unit matrix E. d dt X = AX+XB = d dt X = A E+E B T ) X.4) From these lessons there is no problem to generalize.3) and.4) based on matrices to ones based on any square) matrices or operators on F. Namely, we have { ddt X = AXB = dt d X = A B T ) X, d dt X = AX+XB = dt d X = A I + I B T ) X..5) where I is the identity E matrices) or 1 operators). 3. Quantum damped harmonic oscillator In this section we treat the quantum damped harmonic oscillator. As a general introduction to this topic see [1] or [16] Model Before that we introduce the quantum harmonic oscillator. The Schrödinger equation is given by i h t Ψt) = H Ψt) = ω hn+ 1 ) ) Ψt) by.10) note N = a a). In the following we use t instead of d dt. Now we change from a wave function to a density operator because we want to treat a mixed state, which is a well known technique in Quantum Mechanics or Quantum Optics. If we set ρt) = Ψt) Ψt), then a little algebra gives

9 Quantum Damped Harmonic Oscillator 141 Quantum Damped Harmonic Oscillator 9 i h t ρ = [H, ρ] = [ω hn, ρ] = ρ = i[ωn, ρ]. 3.1) t This is called the quantum Liouville equation. With this form we can treat a mixed state like ρ = ρt) = N u j Ψ j t) Ψ j t) j=1 where u j 0 and N j=1 u j = 1. Note that the general solution of 3.1) is given by ρt) = e iωnt ρ0)e iωnt. We are in a position to state the equation of quantum damped harmonic oscillator by use of 3.1). Definition The equation is given by t ρ = i[ωa a, ρ] µ a aρ+ρa a aρa ) ν ) aa ρ+ρaa a ρa 3.) where µ, ν µ > ν 0) are some real constants depending on the system for example, a damping rate of the cavity mode) 1. Note that the extra term µ a aρ+ρa a aρa ) ν ) aa ρ+ρaa a ρa is called the Lindblad form term). Such a term plays an essential role in Decoherence. 3.. Method of solution First we solve the Lindblad equation : t ρ = µ a aρ+ρa a aρa ) ν ) aa ρ+ρaa a ρa. 3.3) Interesting enough, we can solve this equation completely. 1 The aim of this chapter is not to drive this equation. In fact, its derivation is not easy for non experts, so see for example the original papers [15] and [10], or [1] as a short review paper

10 14 Advances in Quantum Mechanics 10 Quantum Mechanics Let us rewrite 3.3) more conveniently using the number operator N a a where we have used aa = N+ 1. t ρ = µaρa + νa ρa µ+ν Nρ+ρN+ ρ)+ ρ 3.4) From here we use the method developed in Section.. For a matrix X = x ij ) MF) over F x 00 x 01 x 0 x 10 x 11 x 1 X = x 0 x 1 x we correspond to the vector X F dim CF as X = x ij ) X = x 00, x 01, x 0, ; x 10, x 11, x 1, ; x 0, x 1, x, ; ) T 3.5) where T means the transpose. The following formulas ÂXB = A B T ) X, AX+XB) = A 1+1 B T ) X 3.6) hold for A, B, X MF), see.5). Then 3.4) becomes { t ρ = µa a ) T + νa a T µ+ν { = N 1+1 N+ 1 1)+ } 1 1 ρ } 1 1+νa a + µa a µ+ν N 1+1 N+ 1 1) ρ 3.7) where we have used a T = a from the form.15), so that the solution is formally given by ρt) = e t e t{νa a +µa a µ+ν N 1+1 N+1 1)} ρ0). 3.8) In order to use some techniques from Lie algebra we set K 3 = 1 N 1+1 N+ 1 1), K + = a a, K = a a ) K = K + 3.9) then we can show the relations [K 3, K + ] = K +, [K 3, K ] = K, [K +, K ] = K 3.

11 Quantum Damped Harmonic Oscillator 143 Quantum Damped Harmonic Oscillator 11 This is just the su1, 1) algebra. The proof is very easy and is left to readers. The equation 3.8) can be written simply as so we have only to calculate the term ρt) = e t e t{νk ++µk µ+ν)k 3} ρ0), 3.10) e t{νk ++µk µ+ν)k 3 }, 3.11) which is of course not simple. Now the disentangling formula in [4] is helpful in calculating 3.11). If we set {k +, k, k 3 } as k + = ) 0 1, k 0 0 = ) 0 0, k = 1 ) 1 0 ) k 0 1 = k + 3.1) then it is very easy to check the relations [k 3, k + ] = k +, [k 3, k ] = k, [k +, k ] = k 3. That is, {k +, k, k 3 } are generators of the Lie algebra su1, 1). Let us show by SU1, 1) the corresponding Lie group, which is a typical noncompact group. Since SU1, 1) is contained in the special linear group SL; C), we assume that there exists an infinite dimensional unitary representation ρ : SL; C) UF F) group homomorphism) satisfying dρk + ) = K +, dρk ) = K, dρk 3 ) = K 3. From 3.11) some algebra gives e t{νk ++µk µ+ν)k 3 } = e t{νdρk +)+µdρk ) µ+ν)dρk 3 )} = e dρtνk ++µk µ+ν)k 3 )) ) = ρ e tνk ++µk µ+ν)k 3 ) by definition) ρ e ta) 3.13)

12 144 Advances in Quantum Mechanics 1 Quantum Mechanics and we have e ta = e t{νk ++µk µ+ν)k 3} { µ+ν = exp t ν µ µ+ν = ) cosh t )} µ+ν sinh t ) µ sinh t ) ) ν sinh ) t ) cosh t + µ+ν sinh t. The proof is based on the following two facts. ta) = t µ+ν ν µ µ+ν ) = t ) µ+ν µν 0 0 µ+ν ) ) 1 0 ) = t µν 0 1 and e X = 1 n! Xn = 1 n)! Xn + 1 n+1)! Xn+1 X = ta). Note that coshx) = ex + e x = x n n)! and sinhx) = ex e x = x n+1 n+1)!. The remainder is left to readers. The Gauss decomposition formula in SL; C)) ) a b 1 b = d c d 0 1 ) 1d 0 0 d ) ) 1 0 c d 1 ad bc = 1) gives the decomposition

13 Quantum Damped Harmonic Oscillator 145 Quantum Damped Harmonic Oscillator 13 e ta = ν sinh cosh µ+ν t)+ sinh cosh 1 µ+ν t)+ µ cosh t) t) sinh t) 0 ) 0 cosh t 1 0 sinh t) sinh t) 1 t)+ µ+ν + µ+ν sinh t ) and moreover we have ν e ta = exp 0 sinh t) cosh µ+ν t)+ sinh t) 0 0 log cosh µ+ν t)+ sinh exp 0 log 0 0 t)) 0 cosh t)+ µ+ν exp µ sinh t) cosh µ+ν t)+ sinh t) 0 ) ν = exp sinh t ) ) cosh t + µ+ν k sinh + t ) exp log cosh t + µ+ν sinh ) µ exp sinh t ) ) cosh t + µ+ν k sinh. t sinh t)) )) ) t k 3 ) Since ρ is a group homomorphism ρxyz) = ρx)ρy)ρz)) and the formula ρ e Lk) = e Ldρk) k = k +, k 3, k ) holds we obtain

14 146 Advances in Quantum Mechanics 14 Quantum Mechanics ) ν ρ e ta) =exp sinh t ) ) cosh t + µ+ν dρk sinh + ) t ) exp log cosh t + µ+ν )) ) sinh t dρk 3 ) ) µ exp sinh t ) ) cosh t + µ+ν dρk sinh ). t As a result we have the disentangling formula ) ν e t{νk ++µk µ+ν)k 3 } = exp sinh t ) ) cosh t + µ+ν K sinh + t ) exp log cosh t + µ+ν )) ) sinh t K 3 ) µ exp sinh t ) ) cosh t + µ+ν K sinh 3.14) t by 3.13). In the following we set for simplicity ) µ Et) = sinh ) t ) cosh t + µ+ν, sinh t ) Ft) = cosh t + µ+ν ) sinh t, 3.15) ) ν Gt) = sinh ) t ) cosh t + µ+ν. sinh t Readers should be careful of this proof", which is a heuristic method. In fact, it is incomplete because we have assumed a group homomorphism. In order to complete it we want to show a disentangling formula like e t{νk ++µk µ+ν)k 3 } = e ft)k + e gt)k 3 e ht)k

15 Quantum Damped Harmonic Oscillator 147 Quantum Damped Harmonic Oscillator 15 with unknowns ft), gt), ht) satisfying f0) = g0) = h0) = 0. For the purpose we set At) = e t{νk ++µk µ+ν)k 3 }, Bt) = e ft)k + e gt)k 3 e ht)k. For t = 0 we have A0) = B0) = identity and Ȧt) = {νk + + µk µ+ν)k 3 }At). Next, let us calculate Ḃt). By use of the Leibniz rule Ḃt) = f K + )e ft)k + e gt)k 3 e ht)k + e ft)k + ġk 3 )e gt)k 3 e ht)k + e ft)k + e gt)k 3 ḣk )e ht)k { } = f K+ + ġe f K + K 3 e f K + + ḣe f K + e gk 3 K e gk 3 e f K + e ft)k + e gt)k 3 e ht)k { } = f K+ + ġk 3 f K + )+ḣe g K f K 3 + f K + ) Bt) { = f } ġ f + ḣe g f )K + +ġ ḣe g f)k 3 + ḣe g K Bt) where we have used relations e f K + K 3 e f K + = K 3 f K +, e gk 3 K e gk 3 = e g K and e f K + K e f K + = K f K 3 + f K +. The proof is easy. By comparing coefficients of Ȧt) and Ḃt) we have ḣe g = µ, ġ ḣe g f = µ+ν), f ġ f + ḣe g f = ν ḣe g = µ, = ġ µ f = µ+ν), f ġ f + µ f = ν ḣe g = µ, = ġ µ f = µ+ν), f +µ+ν) f µ f = ν. Note that the equation f +µ+ν) f µ f = ν is a famous) Riccati equation. If we can solve the equation then we obtain solutions like f = g = h. Unfortunately, it is not easy. However there is an ansatz for the solution, G, F and E. That is, ft) = Gt), gt) = logft)), ht) = Et) in 3.15). To check these equations is left to readers as a good exercise). From this A0) = B0), Ȧ0) = Ḃ0) = At) = Bt) for all t

16 148 Advances in Quantum Mechanics 16 Quantum Mechanics and we finally obtain the disentangling formula e t{νk ++µk µ+ν)k 3 } = e Gt)K + e logft))k 3 e Et)K 3.16) with 3.15). Therefore 3.8) becomes ρt) = e t exp Gt)a a ) exp logft))n 1+1 N+ 1 1)) expet)a a) ρ0) with 3.15). Some calculation by use of.) gives ρt) = e t Ft) exp Gt)a a T){ exp logft))n) exp logft))n) T} exp Et)a a ) T) ρ0) 3.17) where we have used N T = N and a = a T. By coming back to matrix form by use of 3.6) like exp Et)a a ) T) ρ0) = we finally obtain = m=0 Et) m m! Et) m m! m=0 a a ) T) m ρ0) a m a ) m ) T) ρ0) m=0 Et) m a m ρ0)a ) m m! ρt) = e t Ft) Gt) n a ) [exp n logft))n) n! { m=0 } ] Et) m a m ρ0)a ) m exp logft))n) a n. m! 3.18) This form is very beautiful but complicated! 3.3. General solution Last, we treat the full equation 3.) t ρ = iωa aρ ρa a) µ a aρ+ρa a aρa ) ν ) aa ρ+ρaa a ρa.

17 Quantum Damped Harmonic Oscillator 149 Quantum Damped Harmonic Oscillator 17 From the lesson in the preceding subsection it is easy to rewrite this as { t ρ = iωk 0 + νk + + µk µ+ν)k 3 + } 1 1 in terms of K 0 = N 1 1 N note that N T = N). Then it is easy to see ρ 3.19) [K 0, K + ] = [K 0, K 3 ] = [K 0, K ] = 0 3.0) from 3.9), which is left to readers. That is, K 0 commutes with all {K +, K 3, K }. Therefore ρt) = e iωtk 0 e t{νk ++µk µ+ν)k } ρ0) = e t exp iωtk 0 ) expgt)k + ) exp logft))k 3 ) expet)k ) ρ0) = e t expgt)k + ) exp{ iωtk 0 logft))k 3 }) expet)k ) ρ0), so that the general solution that we are looking for is just given by ρt) = e t Ft) Gt) n a ) n [exp{ iωt logft))}n) n! { m=0 } Et) m a m ρ0)a ) m exp{iωt logft))}n)]a n 3.1) m! by use of 3.17) and 3.18). Particularly, if ν = 0 then from 3.15), so that we have sinh µ Et) = t) cosh µ t) + sinh µ t) = 1 e µt, µ ) µ ) Ft) = cosh t + sinh t = e µ t, Gt) = 0 ρt) = e µ +iω)tn { m=0 1 e µt ) } m a m ρ0)a ) m e µ iω)tn 3.) m! from 3.1). 4. Quantum counterpart In this section we explicitly calculate ρt) for the initial value ρ0) given in the following.

18 150 Advances in Quantum Mechanics 18 Quantum Mechanics 4.1. Case of ρ0) = 0 0 Noting a 0 = 0 0 = 0 a ), this case is very easy and we have ρt) = e t Ft) Gt) n n! a ) n 0 0 a n = e t Ft) Gt) n n n = e t Ft) elog Gt)N 4.1) because the number operator N = a a) is written as N = n n n = N n = n n, see for example.15). To check the last equality of 4.1) is left to readers. Moreover, ρt) can be written as ρt) = 1 Gt))e log Gt)N = e log1 Gt)) e log Gt)N, 4.) see the next subsection. 4.. Case of ρ0) = α α α C) Remind that α is a coherent state given by.16) a α = α α α a = α ᾱ). First of all let us write down the result : { )} ρt) = e α e )t log Gt)+log1 Gt)) exp log Gt) αe +iω)t a + ᾱe iω)t a N with Gt) in 3.15). Here we again meet a term like.3) 4.3) with λ =. αe +iω)t a + ᾱe iω)t a Therefore, 4.3) is just our quantum counterpart of the classical damped harmonic oscillator. The proof is divided into four parts. [First Step] From 3.1) it is easy to see m=0 Et) m a m α α a ) m = m! m=0 Et) m α m α α ᾱ m = m! m=0 Et) α ) m α α = e Et) α α α. m!

19 Quantum Damped Harmonic Oscillator 151 Quantum Damped Harmonic Oscillator 19 [Second Step] From 3.1) we must calculate the term e γn α α e γn = e γn e αa ᾱa 0 0 e αa ᾱa) e γn where γ = iωt logft)) note γ = γ). It is easy to see e γn e αa ᾱa 0 = e γn e αa ᾱa e γn e γn 0 = e eγn αa ᾱa)e γn 0 = e αeγ a ᾱe γa 0 where we have used e γn a e γn = e γ a and e γn ae γn = e γ a. The proof is easy and left to readers. formula.17) two times Therefore, by use of Baker Campbell Hausdorff e αeγ a ᾱe γa 0 = e α e αeγ a e ᾱe γa 0 = e α e αeγ a 0 = e α e α eγ+ γ e αeγ a ᾱe γa 0 = e α 1 eγ+ γ) αe γ and we obtain with γ = iωt logft)). e γn α α e γn = e α 1 e γ+ γ) αe γ αe γ [Third Step] Under two steps above the equation 3.1) becomes ρt) = e t e α Et) 1+e γ+ γ ) Ft) Gt) n a ) n αe γ αe γ a n. n! For simplicity we set z = αe γ and calculate the term ) = Gt) n a ) n z z a n. n! Since z = e z / e za 0 we have

20 15 Advances in Quantum Mechanics 0 Quantum Mechanics ) = e z = e z e za { Gt) n a ) n e za 0 0 e za a n n! } Gt) n a ) n 0 0 a n e za n! } = e z e za { Gt) n n n = e z e za e log Gt)N e za e za by 4.1). Namely, this form is a kind of disentangling formula, so we want to restore an entangling formula. For the purpose we use the disentangling formula e αa +βa+γn = e αβ eγ 1+γ) γ e α eγ 1 γ a e γn e β eγ 1 γ a 4.4) where α, β, γ are usual numbers. The proof is given in the fourth step. From this it is easy to see Therefore u z, v log Gt), w z) e ua e vn e wa = e uwev 1+v)) e v 1) e uv e v 1 a + vw e v 1 a+vn. 4.5) so by noting z 1+log Gt) Gt)) ) = e z e 1 Gt)) log Gt) e Gt) 1 za + log Gt) Gt) 1 za+log Gt)N, z = αe γ = α e iωt Ft) and z = α e γ+ γ = α 1 Ft) we have ρt) = e t Ft) e α Et) 1) e α 1+log Gt) Gt) = e t Ft) e α { 1+log Gt) Gt) Et) 1+ Ft) 1 Gt)) }e Ft) 1 Gt)) log Gt) e Ft)Gt) 1) αe iωt a log Gt) + Ft)Gt) 1) ᾱeiωt a+log Gt)N log Gt) Ft)Gt) 1) αe iωt a log Gt) + Ft)Gt) 1) ᾱeiωt a+log Gt)N. By the way, from 3.15)

21 Quantum Damped Harmonic Oscillator 153 Quantum Damped Harmonic Oscillator 1 Gt) 1 = e t Ft), 1 = e t e t, Et) 1 = Ft)Gt) 1) Ft) and 1 Gt)+log Gt) Ft) Gt) 1) { } = e )t e t + log Gt) Ft) = e t + e )t log Gt) Ft) = Et) 1)+e )t log Gt) we finally obtain { } ρt) = 1 Gt))e α e )t log Gt) log Gt) αe iωt e t a +ᾱe iωt e t a N e = e α e )t log Gt)+log1 Gt)) log Gt) e {αe +iω )t a +ᾱe } iω )t a N. [Fourth Step] In last, let us give the proof to the disentangling formula 4.4) because it is not so popular as far as we know. From 4.4) αa + βa+γn = γa a+αa + βa { = γ a + β ) a+ α ) αβ } γ γ γ = γ a + β ) a+ α ) αβ γ γ γ we have ) a e αa +βa+γn = e αβ γ e γ + β γ = e αβ γ e β γ a e γa a+ α γ ) a+ α γ ) e β γ a = e αβ γ e β γ a e α γ a e γa a e α γ a e β γ a.

22 154 Advances in Quantum Mechanics Quantum Mechanics Then, careful calculation gives the disentangling formula 4.4) N = a a) e αβ γ e β γ a e α γ a e γn e α γ a e β γ a = e αβ γ e αβ γ e α γ a e β γ a e γn e α γ a e β γ a αβ γ = e + αβ γ ) e α γ a e β γ a e γn e α γ a e β γ a αβ γ = e + αβ γ ) e α γ a e γn e β γ eγ α a e γ a e β γ a αβ γ = e + αβ αβ γ)+ γ eγ e α γ a e γn e α γ a e β γ eγ a e β γ a = e αβ eγ 1 γ γ = e αβ eγ 1 γ γ e α γ a e α γ eγ a e γn e β eγ 1 γ a e α eγ 1 γ a e γn e β eγ 1 γ a by use of some commutation relations The proof is simple. For example, e sa e ta = e st e ta e sa, e sa e tn = e tn e seta, e tn e sa = e set a e tn. e sa e ta = e sa e ta e sa e sa = e tesa a e sa e sa = e ta +s) e sa = e st e ta e sa. The remainder is left to readers. We finished the proof. The formula 4.3) is both compact and clear-cut and has not been known as far as we know. See [1] and [16] for some applications. In last, let us present a challenging problem. A squeezed state β β C) is defined as β = e 1 βa ) βa ) ) See for example [4]. For the initial value ρ0) = β β we want to calculate ρt) in 3.1) like in the text. However, we cannot sum up it in a compact form like 4.3) at the present time, so we propose the problem, Problem sum up ρt) in a compact form. 5. Concluding remarks In this chapter we treated the quantum damped harmonic oscillator, and studied mathematical structure of the model, and constructed general solution with any initial condition, and gave a quantum counterpart in the case of taking coherent state as an initial condition. It is in my opinion perfect.

23 Quantum Damped Harmonic Oscillator 155 Quantum Damped Harmonic Oscillator 3 However, readers should pay attention to the fact that this is not a goal but a starting point. Our real target is to construct general theory of Quantum Mechanics with Dissipation. In the papers [7] and [8] see also [13]) we studied a more realistic model entitled Jaynes Cummings Model with Dissipation" and constructed some approximate solutions under any initial condition. In the paper [5] we studied Superluminal Group Velocity of Neutrinos" from the point of view of Quantum Mechanics with Dissipation. Unfortunately, there is no space to introduce them. It is a good challenge for readers to read them carefully and attack the problems. Acknowledgments I would like to thank Ryu Sasaki and Tatsuo Suzuki for their helpful comments and suggestions. Author details Kazuyuki Fujii Address all correspondence to: fujii@yokohama-cu.ac.jp International College of Arts and Sciences, Yokohama City University, Yokohama, Japan References [1] H. -P. Breuer and F. Petruccione : The theory of open quantum systems, Oxford University Press, New York, 00. [] P. Dirac : The Principles of Quantum Mechanics, Fourth Edition, Oxford University Press, This is a bible of Quantum Mechanics. [3] R. Endo, K. Fujii and T. Suzuki : General Solution of the Quantum Damped Harmonic Oscillator, Int. J. Geom. Methods. Mod. Phys, 5 008), 653, arxiv : [quant-ph]. [4] K. Fujii : Introduction to Coherent States and Quantum Information Theory, quant-ph/ This is a kind of lecture note based on my several) talks. [5] K. Fujii : Superluminal Group Velocity of Neutrinos : Review, Development and Problems, Int. J. Geom. Methods Mod. Phys, ), , arxiv: [physics]. [6] K. Fujii and T. Suzuki : General Solution of the Quantum Damped Harmonic Oscillator II : Some Examples, Int. J. Geom. Methods Mod. Phys, 6 009), 5, arxiv : [quant-ph]. [7] K. Fujii and T. Suzuki : An Approximate Solution of the Jaynes Cummings Model with Dissipation, Int. J. Geom. Methods Mod. Phys, 8 011), 1799, arxiv : [math-ph].

24 156 Advances in Quantum Mechanics 4 Quantum Mechanics [8] K. Fujii and T. Suzuki : An Approximate Solution of the Jaynes Cummings Model with Dissipation II : Another Approach, Int. J. Geom. Methods Mod. Phys, 9 01), , arxiv : [math-ph]. [9] K. Fujii and et al ; Treasure Box of Mathematical Sciences in Japanese), Yuseisha, Tokyo, 010. I expect that the book will be translated into English. [10] V. Gorini, A. Kossakowski and E. C. G. Sudarshan ; Completely positive dynamical semigroups of N level systems, J. Math. Phys, ), 81. [11] H. S. Green : Matrix Mechanics, P. Noordhoff Ltd, Groningen, This is my favorite textbook of elementary Quantum Mechanics. [1] K. Hornberger : Introduction to Decoherence Theory, in Theoretical Foundations of Quantum Information", Lecture Notes in Physics, ), 1-76, Springer, Berlin, quant-ph/ [13] E. T. Jaynes and F. W. Cummings : Comparison of Quantum and Semiclassical Radiation Theories with Applications to the Beam Maser, Proc. IEEE, ), 89. [14] J. R. Kauder and Bo-S. Skagerstam : Coherent States Applications in Physics and Mathematical Physics, World Scientific, Singapore, This is a kind of dictionary of coherent states. [15] G. Lindblad ; On the generator of quantum dynamical semigroups, Commun. Math. Phys, ), 119. [16] W. P. Schleich : Quantum Optics in Phase Space, WILEY VCH, Berlin, 001. This book is strongly recommended although it is thick. [17] C. Zachos : Crib Notes on Campbell-Baker-Hausdorff expansions, unpublished, 1999, see I expect that this crib) note will be published in some journal.