On the control by electromagnetic fields of quantum systems with infinite dimensional Hilbert space

Abstract

We analyze the control by electromagnetic fields of quantum systems with infinite dimensional Hilbert space and a discrete spectrum. Based on recent mathematical results, we rigorously show under which conditions such a system can be approximated in a finite dimensional Hilbert space. For a given threshold error, we estimate this finite dimension in terms of the used control field. As illustrative examples, we consider the cases of a rigid rotor and of a harmonic oscillator.

Appendix: Estimate of the dimension of the finite dimensional Hilbert space

We give in the appendix some indications about the way to compute a precise estimate of the dimension of the finite dimensional Hilbert space in the case of a planar rotor.

For some $\epsilon$, we find $N$ according to (14) and we consider the dynamical evolution of the system in the subspace $\mathcal {H}^{(N)}$, which is governed by Eq. (7):

$$\begin{aligned} \frac{1}{i}\frac{d}{d t}|\psi _N(t)\rangle = \left[ H_0^{(N)}+u(t)H_1^{(N)}\right] |\psi _N(t)\rangle . \end{aligned}$$

Assuming that the initial state is $|\phi _1\rangle$, a general solution reads as follows:

$$\begin{aligned} |\psi _N(t)\rangle&= e^{-iH_0^{(N)}t}|\phi _1\rangle \nonumber \\&\qquad +\,\int _0^t e^{-i(t-s)H_0^{(N)}}u(s)H_1^{(N)}|\psi _N(s)\rangle ds. \end{aligned}$$

(16)

Replacing $|\psi _N(t)\rangle$ in the integral term of Eq. (16) by its value given by the same equation (16), we get:

$$\begin{aligned} |\psi _N(t)\rangle&= e^{-iH_0^{(N)}t}|\phi _1\rangle + \int _0^t e^{-i(t-s)H_0^{(N)}}u(s)H_1^{(N)}e^{-iH_0^{(N)}s}|\phi _1\rangle ds \nonumber \\&\quad \!\!+\,\int _0^t\int _0^{s_1} e^{\!-\!i(t\!-\!s_1)H_0^{(N)}}H_1^{(N)} e^{-i(s_1-s_2)H_0^{(N)}}H_1^{(N)}u(s_1)u(s_2)|\psi _N(s_2)\rangle ds_1 ds_2. \end{aligned}$$

(17)

For a fixed number $p\ge 2$, we repeat this operation $p-1$ times, which leads to:

$$\begin{aligned} |\psi _N(t)\rangle&= e^{-iH_0^{(N)}t}|\phi _1\rangle \nonumber \\&\quad +\,\sum _{k=1}^{p-1}\int _{0\le s_k\le s_{k-1} \le \cdots \le s_1\le t} e^{-i(t-s_1)H_0^{(N)}}H_1^{(N)} \nonumber \\&\quad \cdots e^{-iH_0^{(N)}(s_{k-1}-s_k)}u(s_1)u(s_2)\cdots u(s_k)|\phi _1\rangle ds_1ds_2\cdots ds_k \nonumber \\&\quad +\,\int _{0\le s_p\le s_{p-1} \le \cdots \le s_1\le t} e^{-i(t-s_1)H_0^{(N)}}H_1^{(N)} \nonumber \\&\quad \cdots e^{-iH_0^{(N)}(s_{p-1}-s_p)}H_1^{(N)}u(s_1)u(s_2)\cdots u(s_k)|\psi _N(s_k)\rangle ds_1ds_2\cdots ds_k. \end{aligned}$$

(18)

We next compute the projection of this state onto $|\phi _{p+1}\rangle$. Using the tridiagonal structure of the operator $H_1^{(N)}$, one deduces that the first two terms of the right-hand side of Eq. (18) have no contribution. One finally arrives at:

$$\begin{aligned} \langle \phi _{p+1}|\psi _N(t)\rangle =&\int _{0\le s_p\le s_{p-1} \le \cdots \le s_1\le t} \langle \phi _{p+1}|e^{-i(t-s_1)H_0^{(N)}}H_1^{(N)} \nonumber \\&\cdots e^{-iH_0^{(N)}(s_{p-1}-s_p)}H_1^{(N)}\prod _{i=1}^p u(s_i) |\psi _N(s_p)\rangle ds_1ds_2\cdots ds_p, \end{aligned}$$

(19)

where the integrand contains $p$ factors $H_1^{(N)}$. A majorization of this term is given by:

$$\begin{aligned} |\langle \phi _{p+1}|\psi _N(t)\rangle | \le&\int _{0\le s_p\le s_{p-1} \le \cdots \le s_1\le t} \Vert H_1^{(N)} e^{i(s_{p-1}-s_p)H_0^{(N)}}H_1^{(N)} \nonumber \\&\!\cdots e^{iH_0^{(N)}(t-s_1)}|\phi _{l\!+\!1}\!\rangle \Vert \prod _{i=1}^p|u(s_i)|\sqrt{\langle \psi _N(s_p)|\psi _N(s_p)\rangle } ds_1ds_2 \cdots ds_p. \end{aligned}$$

(20)

For the planar rotor, we denote by $c$ the absolute value of the coupling constant. Straightforward computations give

$$\begin{aligned} \sup _{s_1,\cdots ,s_p} \left\| H_1^{(N)}e^{i(s_{p-1}-s_p)H_0^{(N)}}H_1^{(N)}\cdots e^{iH_0^{(N)}(t-s_1)}|\phi _{p+1}\rangle \right\| \le 2^pc^p. \end{aligned}$$

(21)

Indeed, the product $H|\psi \rangle$, with $H$ tridiagonal, can always be written as $H|\psi \rangle = a|\psi _1\rangle + b|\psi _2\rangle + c|\psi _3\rangle$, where $|\langle \psi _i |\psi _i\rangle |\le | \langle \psi |\psi \rangle |$ and $(a,b,c)$ are the maxima of the nonzero diagonals. In our case, one of the diagonal is void so $\Vert H |\psi \rangle \Vert \le 2c$. We can also show by induction that

$$\begin{aligned} \int _{0\le s_p\le s_{p-1} \le \cdots \le s_1\le t}\prod _{i=1}^p |u(s_i)|ds_1\cdots ds_p=\frac{1}{p!}\left( \int _0^t|u(s)|ds\right) ^p, \end{aligned}$$

(22)

which leads to

$$\begin{aligned} |\langle \phi _{p+1}|\psi _N(t)\rangle |\le 2^pc^p \frac{K^p}{p !}. \end{aligned}$$

(23)

The estimate (23) is valid in $\mathcal {H}^{(N)}$. If one considers the actual solution of the infinite dimensional system in $\mathcal {H}$, one yields, for $\epsilon$ and $N$ chosen according to (14),

$$\begin{aligned} |\langle \phi _{p+1}|\psi (t)\rangle |\le 2^pc^p \frac{K^p}{p !}+\varepsilon . \end{aligned}$$

(24)

Estimate (24) is correct for every $\varepsilon >0$. Letting $\varepsilon$ goes to zero, one finally gets

$$\begin{aligned} |\langle \phi _{p+1}|\psi (t)\rangle |\le 2^pc^p \frac{K^p}{p !} \end{aligned}$$

(25)

Estimate (25) can in turn be used to refine the condition (14). Since $c=1/2$, we obtain:

$$\begin{aligned} |\langle \phi _{p+1}|\psi (t)\rangle |\le \frac{K^p}{p !}, \end{aligned}$$

(26)

which gives an estimate of the probability of transitions outside the subspace $\mathcal {H}^{(p)}$. If $K\le (\varepsilon p !)^{(1/p)}$, then this probability is lower than $\varepsilon$. In addition, in the case $K^{p+1}<2\varepsilon p!$, we get $\frac{K^p}{p!}<\frac{2\varepsilon }{K}$ and

$$\begin{aligned} \int _0^t \Vert u(t)P^{(p)}H_1|\tilde{\psi }(\tau )\rangle d\tau \Vert \le \frac{2\varepsilon }{K}\frac{K}{2}, \end{aligned}$$

(27)

i.e. the dynamics in the finite subspace $\mathcal {H}^{(p)}$ is close to $\varepsilon$ to the exact dynamics.