4.4 Conditional Quantum-state Preparation
4.4.3 Preparation of an Arbitrary State
in order to obtain a maximized success probability of
Pn=√ 827
4e3
|h−β|ni|3
p1− |h−β|ni|2. (4.89)
For eachn, the maximum ofPn is reached atβ=√
n. In Fig. 4.9, we plotPn for a range of β, for = 0.1, or a state overlap of≥90%. We can see that the probability of producing |˜ni decreases rather quickly asnincreases.
This dependence (4.89) onβ comes from two sources, which we can understand better by going to the phase-space reference frame centered at the equilibrium position of the oscillator when the photon is inside the cavity. In this reference frame, the complex amplitude of the coherent states being superimposed are located on a circle with distance β away from the center, while the target we would like to prepare is simply the Fock state|ni. Although the photon’s wave function selects out an oscillator state proportional to|ni, this post-selection does not improve the intrinsic overlap between all those that participate the superposition, which is actually proportional to
|h−βeiφ|ni|2=|h−β|ni|2. (4.90)
This explains the dependence ofmhψ|ψimon beta. The other factor of dependence onβis that when the target state has a very low overlap with the individual members|βeiφiof the superposition, the requirement on the accuracy of photon arrival time, or ∆τ, increases, as shown in Eq. (4.86).
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P0
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PH1
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0.2 0.5 1 2 5
10-4 10-3 10-2 10-1
Β
P
Figure 4.9: Minimum success probability for states in Hilbert spaces H1,2,...7 (solid curves with markers), together with success probability for producing single displaced Fock states, P0,1,2,...,7
(dashed curves without markers). Fidelity is fixed at 10%. Note thatP0would become greater than 1 at low values ofβ—but in this case our approximation in obtaining ∆τ breaks down.
This is an additional periodic modulation (with period 2π/ωm) of the photon’s wave function. We caution that in order for the summation in Eq. (4.92) to converge, ifcn does not go to zero for all n≥N, then it must decay very fast whenn→+∞, due to the presence of the√
n! factor (which grows faster thanβ−n).
As in the previous subsection, we obtain the conditional state at τ ≡ωmt= 2π,4π, . . ., as well as anyτthat is substantially large. Again, let us considerτ= 2π, this gives the conditional state of
|ψim=πγ3/2e−ωmπγe2πiβ2
ωmZ |ψtgi. (4.95)
We can use the same approach as the previous subsection to evaluate the probability with which this conditional state is achieved with a high overlap. For a minimum overlap of 1−, we require
|2π−τ| ≤∆τ =
√8π
P+∞
m=0˜cm
p1− |h−β|ψtgi|2
. (4.96)
Note that this ∆τ diverges if P+∞
m=0c˜m = 0, because in this case the overlap does not vary at O[(τ−2π)2] order. Assuming the target state to be generic, then the probability for obtaining this state is then
P|ψi= 2√ 8
πγ
ωm
3 e−2πγωm
1−
P+∞
n=0h−β|ni2˜cn
2−1/2
P+∞
m=0˜cm
+∞
X
j,k=0
˜ cjc˜∗k 1 +i(j−k)ωγ m
. (4.97)
Here the choice ofγ/ωmdepends on the target quantum state, but if we assume this dependence is
weaker than the pre-factor, and continue to use Eq. (4.88), then we obtain
P|ψi= 27 e3
r 2
1−
P+∞
n=0h−β|ni2c˜n
2−1/2
P+∞
m=0˜cm
+∞
X
j,k=0
˜ cj˜c∗k 1 + 2πi(j−k)3
. (4.98)
As it turns out,P|ψidepends on the detail of|ψi—even if we only try to create a combination of
|˜0iand|˜1i, the combination coefficients would lead to very different success probabilities. In order to provide a concrete measure of the ability of our state-preparation scheme, we have chosen to compute the minimum success probabilities of creating all the states in the mechanical oscillator’s Hilbert subspaces spanned by the lowest displaced Fock states, e.g.,H1≡Sp{|˜0i,|˜1i},H2≡Sp{|˜0i,|˜1i,|˜2i}, etc. We define
PHj = min
|ψi∈Hj
P|ψi, Hj = ( j
X
l=0
αl|˜li:αl∈C )
. (4.99)
In Fig. 4.9, we plot PH1, PH2, . . . , PH7 as functions of β (in solid purple curves). Because H1⊂ H2⊂ · · · ⊂ H7, it is increasingly difficult to create all states inHj with higher values ofj, and thereforePH1 ≥PH2≥. . . PH7, namely our success probability decreases globally whenjincreases.
In fact, as we overlay the single-Fock-state success probabilities P0, P1, . . . , P5, we also discover that for anyPHj(β), it asymptotes toP0at higherβ, and toPj at lowerβ; moreover, the transition between these two asymptotic regions are brief, and the PHj(β) curves do not lie much below the minimum ofP0 andPj.
This asymptotic behavior can be understood from the behavior ofPn, the success probability for single (displaced) Fock states. For smallerβ, it is much more difficult to prepare a higher Fock state, therefore, ifβ is sufficiently small, the difficulty of preparingHj is dominated by the preparation of |˜ji, the single most difficult state in the space to prepare—and therefore PHj agrees with Pj. Vice versa, for sufficiently largeβ, the difficulty of preparingHj lies in the preparation of ˜0i, and therefore PHj would agree with P0. The fast transition between the two extremes indicates that when trying to prepare states in Hj, the difficulty either lies in|˜0i, or in |˜ji, and only for a small region ofβthe two difficulties might compete with each other—while none of the intermediate states contribute to the difficulty of state preparation. This is consistent with the relative locations of the Pncurves in Fig. 4.9: (i) for anyβ,P1,2,...,j−1are always much greater than the minimum ofP0and Pj, and (ii) as we move away from theβ at which P0 andPj crosses each other, their discrepancy increases quickly.
As a matter of practicality, we see that if we chooseβ ≈0.87 the probability of achieving, with an overlap (or fidelity) above 90%, any superposition of|˜0iand|˜1i(i.e., any member of the subspace H1) is guaranteed to be above 6.3%. On the other end, with a probability of at least 0.1%, we can
produce all states in the 8-dimensional subspace ofH7.