Quantum stochastic differential equations

27. Quantum stochastic differential equations in the sense of Hudson and Parthasarathy; Existence and Uniqueness of solutions.

jt(X) = X + f js(eab(X))dAba(8)ds Jo

To solve this, we construct the iterative scheme:

jt(n+1X) = X + f > 0

Jo

Then

< w(«)b(n+1) w-jt(nV)i./w(W) >

= [ < y^(w)|(jin)(^(X))-j<”-1)(^(X))|y^(W) > ub(s)ua(s)ds Jo

from which the desired convergence results are derived. Specifically, we define

An+1(i) =< ¿(u)|jt(n+1)(X) - j(tnX)f ® 0(u) >

Then, we get

An+i(t) < I An(s) || u(s) ||2 ds

Jo

which by application of Gronwall’s inequality yields

An+1(t) < C( f || u(s) || ds)n/n

Jo

from which existence of a solution to the qsde is easily established.

Some practical applications of quantum probability

Computation of scattering cross sections

A. B are two Hamiltonians, B=A+V. The wave operators are

Q+ = limt^xexp(itB) ,exp(—itA),

Q_ = limt_i_ooexp(itB).exp(—itA)

The scattering matrix is

s =

Let |i > be the input free state at energy E. Then the output free state at the same energy E is |o >= S(E)i >. The probability of scattering within a cone C is given by

II XcS^i ||2

Further, the probability current density after scattering is given by

J = (ie/2m)(poVd>* - o)

and we can compute using this, the probability that per unit time particles scattered into a solid angle Qq at infinite radius as

/ J(r).r.r2dQ(r)

•' Qofrom which the scattering cross section is easily evaluated by taking the incident state (j>i as corresponding to a single particle moving per unit area per unit time at a perpendicular distance between [6, b + db] from the scattering centre. In this analysis, we take A = — /“2 m and V = V(r) to be the potential of the

scattering centre. The standard reference for such problems is Werner Amrein, ” Hilbert space methods in quantum mechanics”, CRC press.

The wave function of our universe

Wheeler-Dewitt-Hartle-Hawking’s theory. The Hartle-Hawking theory of the probability distribut ion of the radius of our universe with a scalar field coupling.

Computation of tunneling probabilities of a quantum particle through a potential barrier with applications to tunneling diode

Computations based on quasi-classical quantum mechanics. This analysis is also applicable to some quantum field theoretic problems like the quantized KG field perturbed by a Higgs potential. Such a problem in a finite region, say a finite box can be regarded using standard Fourier series analysis as a infinite sequence of 3-D harmonic oscillators with mutual interaction generated by the Higgs potential.

Probability of induced transitions in a laser

The interaction of Light with atoms- The Glauber-Sudarshan theory.

Energy bands in a semiconductor

the theory based on Bloch wave functions; Specialization to the Kronig-Penney model.

Modeling quantum systems coupled to a noisy bath

—The Hudson-Parthasarathy equation and its solution using Maasen’s kernel approach and using the functional form of the Glauber-Sudarshan representation.

= (-(iff + P)dt + LrdA + L2dA* + SdA)ff(i) = dW(t).ff(t)

The joint state of system and bath is

p(t) =

which satisfies

dp(t) = dU(t)p(0)U(ty + U(t)p(0)dU(ty + dU(t)p(0)dU(ty

= dW(t)p(t) + p(t)dW(t)* + L1P(t).L*dt + SdAp(t)S* + Sp(t)dA*L*

+LidArho(t)S*

Note that

dW.p + p.dW* =

—i[H, p]dt — {P, p}dt + LdAp+ L2dA* p + p.dA’iy + p.dA.L*2 + (SdAp + dApS*

To solve this Schrodinger equation for the joint state p of system and bath, we assume

p(t) = f F(t, u, u) ® |e(u) >< e(u)|dw.du

where

du = Tit>odu(t), du = IIt>odu(t)

are path measures. Note that u 6 L2(R+) which is isomorphic to /2(Z+). We observe that p(t) by definition commutes with dA(t), dA(t)* and <7A(i) provided that we assume that the initial bath state commutes with these operators and this along with quantum Ito’s formula has been used in the derivation of the above relations. If we do not make such an assumption, then we would get

dp(t) = dW.p + p.dW* - i[H,p]dt - {P.p}dt + LxdA.p.dA*L + LYdA.p.dA.L*2

+L1dA.p.dA.S* + L2dA*.p.dA*.iy + L2dA* .p.dA.L*2 + L2dA* .p.dA.S*

+SdA.p.dA*iy + SdA.p.dA.L*2 + SdA.p.dA.S*

Now,

dAp.dA* = f F(t,u,u)dA(t)e(u) >< e(u)|dudu = 0(dt2) and hence this term can be neglected. Note that

< e(w)|2dt2|e(u) >< e(w)|

Further,

< e(u)| = u(t)dt.e(u) >< e(u)|

cL4(t)*|e(u)

< e(u)|cL4(t)* = u(t)dt|e(u) >< e(w)| = ^|e(u + ex[t,t+di

32

dA(t)e(u) >< e(u)dA(t) = XiO+dt]^-^ dA(t)e(u) >< e(u)dA(t) = u(t)dt.x[tit+dt]j^{ which can be neglected.

Casting the HP equation in functional derivative form

dU(t) = (-(¡H + P)dt + L^dA + L-2dA* + SdA)U(t)

Let

So,

U(t) = / F(t,u, u) ® |e(w) >< e(u)dudu

LidA(t)U(t) = f(LiF(t,u,u)) ® < e(u)dudu

= dt [ u(^t')L1F(t, u, u) ® |e(w) >< e(u)dudü

L2dA(t)* = fL2F(t,u,u) ® cM(t)*|e(u) >< e(u)dudu

L2F(t,u,u) ® . le(tt) >< e(u)dudu

ou(t)

/r 8F(t,u,u), , . . .. ,

L2—e je «) >< e(u)dudu

6u(t)

Also

SdA.U(t)= / SF(t, u, u) ® dA|e(w)

>< e(u)dudu

SF(t, u, u) <8> dt 1dA*dA|e(ii) >< e(u)dudu

u(t)SF(t,u,u) ® dA*e(u) >< e(u)dudu

= dt u(t)SF(t,u,u)

<5 i , , . , . e(tt) bu(ty K

>< e(u)ldudu

. . (u(t)F(t, u,«)) ® lefu) >< e(u)dudü du(t)

Thus, the HP Schrodinger equation translates to the following functional differ

ential equation:

dF(t.u.u) , ,rr

---Tpj—— = (—iH + P)F(t,u,u)

.. 8F(t,u,u) 8{u(t)F(t,u,u))

 
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