JP5367663B2 - Quantum information processing method and quantum information processing apparatus - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a waveform of an optical pulse most suitable for execution of population transfer based on STIRAP. <P>SOLUTION: According to one embodiment of the invention, a quantum information processing method and apparatus transfers a state of a physical system having two spin-down states ¾0&gt; and ¾1&gt; and one spin-down state ¾2&gt;, from ¾0&gt; to ¾1&gt; in a transfer time T by irradiating the physical system with two pulses (hereafter, a pulse 0 and a pulse 1) resonating with transition from ¾0&gt; to ¾2&gt; and transition from ¾1&gt; to ¾2&gt;. The pulses are temporally changed sufficiently more slowly than their Rabi frequencies and a quantum state is adiabatically changed in a center portion of the transfer time, and the quantum state is non-adiabatically changed properly in start and end portions of the transfer time, so that the probability of the existence of the spin-up state ¾2&gt; in the adiabatic change in the center portion can be minimized. <P>COPYRIGHT: (C)2012,JPO&amp;INPIT

Description

Embodiments of the present invention, quantum information processing apparatus and quantum information processing method, a method of particular quantum state of matter by light manipulation.

  As a typical method for manipulating the quantum state of a physical system such as an atom with light, there is a method called STIRAP (stimulated Raman adiabatic passage).

  Since STIRAP is based on the state manipulation method called adiabatic passage, there is an advantage that it does not depend on the details of the waveform of the optical pulse used. In addition, since an energy eigenstate that does not include an excited state (called a dark state) is used, there is an advantage that it is less susceptible to decoherence due to relaxation of the excited state.

  In STIRAP, the existence probability of an excited state can be reduced as much as possible by ideally taking an infinitely long time or using infinitely strong light. However, in practice, since population movement is performed using light of a finite power within a finite time, the existence probability of the excited state is not zero, and the influence of the relaxation cannot be completely ignored. The existence probability of this excited state depends on the waveform of the optical pulse. Therefore, if it is desired to minimize the influence of relaxation of the excited state, the waveform of the optical pulse to be used must be devised.

JP 2005-134761 A

K. Bergmann, H. Theuer, B. W. Shore, Rev. Mod. Phys. 70, 1003 (1998). H. Goto, K. Ichimura, Phys. Rev. A 70, 012305 (2004).

  The problem to be solved by the invention is to provide an optical pulse waveform that is optimal for performing population movement by STIRAP.

を満たすことを特徴とする。 A quantum information processing method and apparatus according to an embodiment of the present invention provides a | 0>-| 2> transition, | 1> for a physical system having a lower two state | 0>, | 1>, and an upper one state | 2>. Quantum that changes the state of the physical system from | 0> to | 1> during the movement time T by irradiating two pulses (hereinafter, pulse 0 and pulse 1 respectively) that resonate with the transition. Information processing method, etc.

It is characterized by satisfying.

の例を表す図。
μ₀＝μ₁の場合の本発明の

の例を表す図。
μ₀＝μ₁の場合の本発明の励起状態の確率と非ダーク状態の確率の例を表す図。 μ₀≠μ₁の場合の本発明と従来例の比較を表す図。 実施例１で用いられる実験系を説明する図。 Pr³⁺:Y₂SiO₅のPr³⁺イオンのエネルギー準位とレーザーの周波数設定を説明する図。 Conceptual diagram of STIRAP. The figure which shows the example of the rabbi frequency of the optical pulse for STIRAP. The figure which shows the calculation result of the probability amplitude of each state in STIRAP using the optical pulse of FIG. The figure showing the comparison of this invention and a prior art example in the case of (micro | micron | mu) ₀ = micro ₁ . of the present invention when μ ₀ = μ ₁

FIG.
of the present invention when μ ₀ = μ ₁

FIG.
diagram illustrating an example of the probability of the probability and a non-dark state of the excited state of the present invention in the case of μ ₀ = μ _1. diagram showing the comparison of the present invention and the conventional example when μ ₀ ≠ μ _1. 2 is a diagram illustrating an experimental system used in Example 1. FIG. Pr ^3+: Y ₂ energy levels and laser diagram illustrating the frequency setting of the Pr ³⁺ ions of SiO _5.

  First, the principle of population movement by STIRAP will be described.

Consider a Λ-type three-level system as shown in FIG. Assume that the starting state is | 0> and that the state is shifted to | 1> by STIRAP. For this purpose, an optical pulse resonating with the | 0>-| 2> transition and | 1>-| 2> transition is irradiated. The Rabi frequency of these light pulses is set as shown in FIG. First, an optical pulse that resonates with the | 1>-| 2> transition is emitted, and an optical pulse that resonates with the | 0>-| 2> transition is emitted along the way. (In this way, the pulse order in which the pulse that resonates in a state where there is no population at the beginning is first called a counterintuitive order is one of the features of STIRAP.)
The Hamiltonian of this system and its eigenvalues and eigenstates are as follows:

  In the process using two optical pulses as described above (STIRAP), the dark state initially matches the initial state | 0>, and then the dark state changes from | 0> to | 1>. If the time variation of the electric field amplitude of the light pulse is sufficiently slow, the state of the system hardly changes to another eigenstate (adiabatic process), follows the dark state, and the state of the system also changes from | 0> to | 1>. Change (Adiabatic passage). This is population movement by STIRAP.

One advantage of STIRAP is that it does not depend on the details of the optical pulse waveform utilized. The conditions of the optical pulse for STIRAP to succeed are that the dark state changes from | 0> to | 1>, and that the amplitude change is sufficiently slow. May be used. (Figure 2 shows an example of a Gaussian pulse.)
Another advantage of STIRAP is that it is less susceptible to excited state relaxation. During STIRAP, the state of the system is always very close to the dark state. Since the dark state does not include an excited state, the influence of relaxation of the excited state is small.

However, as long as STIRAP is performed for a finite time and a finite light intensity, an excited state actually exists with a certain probability. FIG. 3 shows the STIRAP simulation results when the Gaussian pulse of FIG. 2 is used (the vertical axis is the probability amplitude). The state of the system has changed from | 0> to | 1> as expected, but it can be seen that the existence probability of the excited state increases along the way. When a relaxation rate is applied to a value I _e obtained by integrating the existence probability P _e (t) of the excited state with the movement time, the probability that the excited state is relaxed within the movement time is obtained. Since population movement ends in failure when relaxation occurs, this relaxation probability can be considered as failure probability of population movement. Thus, STIRAP can reduce the effect of relaxation of the excited state, but the relaxation probability cannot be reduced to zero, and the transfer efficiency decreases accordingly.

The existence probability P _e (t) of the excited state depends on the waveform of the optical pulse to be used. Therefore, in order to minimize the relaxation probability of the excited state and maximize the efficiency of population movement, it is necessary to devise the waveform of the optical pulse. The present invention provides the best waveform for this efficiency maximization.

  In order to clarify the problem, we can paraphrase the problem we are thinking of as the following optimization problem.

In population movement by STIRAP, when the total optical power I _in , the movement time T, and the transition dipole moment μ ₀ , μ _{1 of} each transition are given, the existence probability P _e (t) of the excited state is integrated with the movement time. Find the pulse waveform that minimizes the measured value I _e .

  This optimization problem minimizes the relaxation probability of the excited state (population transfer error probability) with the total optical power and travel time as constraints, but from a different perspective, the total optical power and error probability are constraints. If so, the optimization is such that the travel time is the shortest, and the power efficiency is maximum if the travel time and the error probability are constrained.

  This optimization problem is handled mathematically as follows.

を得る。 The state of the system at time t is expressed using the energy eigenstate

Get.

In STIRAP, the initial state is equal to the dark state, so the initial condition is

Also, during STIRAP, the probability of being in a state other than the dark state (hereinafter referred to as the “non-dark state”) and the probability of being in the excited state are almost equal (under conditions suitable for the current problem) (details) Can be approximated by the probability of being in a non-dark state. From the above approximation, the existence probability P _e (t) of the excited state is

となる。この場合に上記の最適化問題を書き直すと、次のようになる。

It is difficult to find a general solution for this optimization problem. Therefore, first consider the case of μ ₀ = μ ₁ . (If μ ₀ ≈μ ₁ in fact, the solution obtained here can be used as it is. The optimum solution when μ ₀ ≠ μ ₁ is the optimum solution when μ ₀ = μ _{1 obtained} here. (This will be obtained by extending the features of.)
Since it is better to use all the optical power without waste, let us consider a case where the equal sign holds in inequality (4). Furthermore, when μ ₀ = μ ₁ , the sum of the squares of the rabbi frequencies of the two optical pulses is constant,

It becomes. In this case, the above optimization problem is rewritten as follows.

The solution obtained here is discontinuous at the two ends, but the Schrodinger equation is not affected by this discontinuity.

では99.9999%以上。また、ガウシアンパルスはパワーの拘束条件である不等式（４）を満たしながら最適なパラメータを探すのが難しいため、２つのパルスのラビ周波数のピーク値がともに最大パワーの場合の値であるとした。不等式（４）をきちんと考えると、従来例は図４の結果よりも悪くなる。）Tが十分長いとき、最適解のI_eは従来例の約半分に抑えられていることがわかる。これは速さにして２倍速くなることに相当する。 Both ends have a function form with an exponential function added. (The exponential function can be ignored except at both ends.)
FIG. 4 shows the effect of the optimal solution when μ ₀ = μ ₁ . As a conventional example to be compared, two Gaussian pulses having the same width and pulse width selected to be optimum values were used. (Movement efficiency was limited to 99.99% or more, and the one with the smallest I _e was selected. The movement efficiency of the optimal solution is

Then 99.9999% or more. In addition, since it is difficult for the Gaussian pulse to find the optimum parameter while satisfying the inequality (4) which is a power constraint condition, it is assumed that the peak values of the Rabi frequency of the two pulses are both the maximum power. Considering inequality (4) properly, the conventional example is worse than the result of FIG. ) It can be seen that when T is sufficiently long, I _e of the optimal solution is suppressed to about half that of the conventional example. This corresponds to a speed increase of 2 times.

中央では一定で、両端では大きく（指数関数的に）変化する。また、通常のSTIRAPでは、

このように、最適解は、通常のSTIRAPからは到底予想できない特徴を持っている。
Here, the feature of the optimum solution when μ ₀ = μ ₁ will be described.

It is constant at the center and varies greatly (exponentially) at both ends. In addition, in normal STIRAP,

Thus, the optimal solution has characteristics that cannot be predicted from ordinary STIRAP.

In the following, we explain that the strange feature of this optimal solution is cleverly used to minimize I _e , and clarify its physical meaning.

の場合の励起状態にいる確率と非ダーク状態にいる確率の計算結果を示す。この初めの部分を見ると、非ダーク状態はブライト状態から励起状態へと素早く変化している。これは、初めの

が負から正へと素早く変わる部分が、非ダーク状態をブライト状態から励起状態へと変化させる役割をしていることを意味する。これがI_eを最小にするのに重要な役割を果たしていることを説明するために、前述の確率振幅の近似解を、始状態が非ダーク状態を含む場合に拡張したものを考察する。（以下の式では、STIRAPの開始時刻０と区別するため、初期時刻をt₀とする。これは時刻０に近く、t₀<<T。）その場合、

のときに最小となる。これは、式（１）からわかるように、時刻t₀での状態の非ダーク状態が励起状態のみである場合である。このように、I_eは初期状態に依存し、I_eを最小にする初期状態は非ダーク状態がブライト状態を含まず励起状態のみで式（８）のようになる場合である。 FIG.

The calculation results of the probability of being in an excited state and the probability of being in a non-dark state are shown in FIG. Looking at this first part, the non-dark state is quickly changing from the bright state to the excited state. This is the first

The part that changes quickly from negative to positive means that the non-dark state changes from the bright state to the excited state. In order to explain that this plays an important role in minimizing I _e , consider an extension of the above-described approximate solution of the probability amplitude to the case where the starting state includes a non-dark state. (In the following equation, the initial time is t ₀ to distinguish it from the start time 0 of STIRAP. This is close to time 0, t ₀ << T.)

At the minimum. This is a case where the non-dark state of the state at time t ₀ is only the excited state, as can be seen from Equation (1). As described above, I _e depends on the initial state, and the initial state where I _e is minimized is a case where the non-dark state does not include the bright state and only the excited state is represented by Equation (8).

が負から正へと素早く変わる部分は、初めの短い時間t₀で（非断熱的な過程によって）この適切な始状態（確率振幅が式（８）となる状態）の準備しているである。これによって、通常のダーク状態が始状態であるSTIRAPに比べてI_eを小さくできる。この状態準備を成功させるために、通常のSTIRAPとは異なり、

となる。これは、式（１）からわかるように、非ダーク状態は常に励起状態であることを意味し、非ダーク状態と励起状態を同一視できることを意味する。これが前述の「非ダーク状態の確率と励起状態にいる確率はほぼ等しくなる」の根拠である。この前提条件は、始状態が式（８）の適切な状態に準備されていることであり、そうでなければこれは成り立たない。しかし、この状態準備をせず通常のように始状態がダーク状態の場合は、上記のようにI_eがおよそ２倍になってしまうため、適切な始状態の場合よりも悪くなると予想される（良くなったとしても大きな改善は期待できない）。 At the beginning of the optimal solution

The part where changes quickly from negative to positive is the preparation of this appropriate initial state (with probability amplitude of equation (8)) in the first short time t ₀ (by a non-adiabatic process). . As a result, I _e can be made smaller than STIRAP where the normal dark state is the starting state. Unlike normal STIRAP, to make this state preparation successful,

It becomes. As can be seen from equation (1), this means that the non-dark state is always an excited state, and the non-dark state and the excited state can be identified. This is the basis for the aforementioned “probability of non-dark state and probability of being in an excited state”. This precondition is that the starting state is set to the appropriate state of equation (8), otherwise this is not true. However, if this state is not prepared and the starting state is a dark state as usual, I _e is approximately doubled as described above, so it is expected to be worse than the appropriate starting state. (Even if it gets better, we can't expect a big improvement).

初めの部分と同様の非断熱的な過程によって効率を１にする（状態を完全に|1>にする）のが、この最後の部分の役割である。（時間が短いため、ここの部分のI_eへの影響は無視できる。）この非断熱過程をうまく行うために、この最後の部分では、通常のSTIRAPとは異なり、

If an appropriate start state is prepared in the first short time t ₀ ,

The role of this last part is to make the efficiency 1 (by making the state completely | 1>) by a non-adiabatic process similar to the first part. (Because of the short time, the influence of this part on I _e is negligible.) In order to successfully perform this non-adiabatic process, this last part differs from normal STIRAP,

In summary, the features and roles of the optimal solution when μ ₀ = μ ₁ are as follows.

In the first short time, a non-adiabatic process changes the non-dark state from the bright state to the excited state with an appropriate probability amplitude. Therefore, unlike normal STIRAP

In this way, the optimal solution simultaneously realizes perfect movement efficiency and minimum I _e by a clever pulse waveform that cannot be imagined from ordinary STIRAP.

Hereinafter, the case where μ ₀ ≠ μ ₁ is considered. The feature of the optimal solution when μ ₀ = μ ₁ is extended to the case of μ ₀ ≠ μ ₁ .

である。これは、式（８）の自然な拡張になっている。（μ₀＝μ₁の場合と同様、時刻t₀における非ダーク状態は、励起状態のみでブライト状態を含まない。）よって、μ₀＝μ₁の場合と同様の非断熱過程によって、初めの短い時間t₀で式（９）を満たすように非ダーク状態をブライト状態から励起状態へ変化させればよい。

An appropriate state is prepared in the first short time t ₀ , and the integrated value of the probability of the subsequent excited state (approximate non-dark state probability) is minimized. The integral value is approximately as follows.

It is. This is a natural extension of equation (8). (As in the case of μ ₀ = μ ₁ , the non-dark state at time t ₀ is only the excited state and does not include the bright state.) Therefore, by the same non-adiabatic process as in the case of μ ₀ = μ ₁ , What is necessary is just to change a non-dark state from a bright state to an excited state so that Formula (9) may be satisfy | _filled in short time t0.

(For example, the one with the minimum I _e is selected when the movement efficiency is 99.99% or more.) In this way, the function form in the central portion excluding the short time at both ends of the movement time is determined.

を正から負へ変化させる。これも初めの非断熱過程と同様に、μ₀＝μ₁の場合の関数形である式（７）を利用する。それが成功するように、

は最後のその非断熱過程を行う際にほぼ一定であるようにする。この最後の非断熱過程にかかる時間は、

Finally, in order to change the state completely to | 1> by a non-adiabatic process,

Is changed from positive to negative. As in the first non-adiabatic process, this also uses Equation (7), which is a function form when μ ₀ = μ ₁ . As it succeeds,

Makes it almost constant during the last non-adiabatic process. The time required for this last non-adiabatic process is

The effect of this optimal solution was confirmed by simulation. The result is shown in FIG. As a conventional example, Gaussian pulses having the same width were used. (Similar to Fig. 4, the movement efficiency was limited to 99.99% or more, and the width and pulse interval where I _e was minimized were selected. In addition, the Gaussian pulse had the maximum power of both peak values of the rabbi frequency of the two pulses. If the inequality (4) is considered properly, the conventional example is worse than the result of FIG. 8. As can be seen from FIG. Double improvement was obtained.

  Finally, the characteristics of general optimal solutions are summarized.

以上が、本発明のパルス波形の特徴である。 In the first short time, the non-dark state is changed from the bright state to the excited state with an appropriate probability amplitude (Equation (9)) by a non-adiabatic process.

The above is the feature of the pulse waveform of the present invention.

  The role of each feature is as follows.

  The first feature is the greatest feature of the present invention. This role is to reduce the relaxation probability compared to normal STIRAP starting from the dark state by first preparing the appropriate state by a non-adiabatic process. In order to perform a non-adiabatic process to prepare this state, the first part does not satisfy the usual STIRAP conditions. Therefore, this feature cannot be imagined from ordinary STIRAP.

The role of the second feature is to minimize the time integration of the existence probability of the excited state in the central portion excluding both ends of the moving time, and as a result, minimize the relaxation probability.

  The role of the third feature is to make the transfer efficiency 1 (by making the state completely | 1>) by a non-adiabatic process similar to the first part. For this reason, as in the first part, the last part does not satisfy the normal STIRAP conditions. Therefore, this feature, like the first one, cannot be imagined from ordinary STIRAP.

  In the present invention, it is important for the effect to specify the function form, but even a slightly different function form is a function form that is close to the function form of the present invention as seen by those skilled in the art, and the effect. Is equivalent to the present invention from the viewpoint of a person skilled in the art, the function form is essentially the same as the function form of the present invention, and even if such a function form is used, it falls within the scope of the present invention. To do.

  As described above, according to the embodiment of the present invention, it is possible to minimize the relaxation probability of the excited state in population movement by STIRAP performed within a certain period of time using a light source having a certain intensity.

  Examples of the present invention will be described below.

Example 1
STIRAP in a solid was first realized using the Pr ³⁺ ion state of Pr ³⁺ : Y ₂ SiO ₅ (hereinafter Pr: YSO) (H. Goto and K. Ichimura, Phys. Rev. A 74). , 053410 (2006)). In this embodiment, Pr: YSO is used as a target. (It can be similarly realized for laser-cooled atoms and ions.) Although the Gaussian pulse is used in the above paper, the pulse waveform of the present invention is used in this embodiment.

An experimental system for this example is shown in FIG. The light source is a frequency stabilized CW ring dye laser. As in the above paper, a laser beam with three frequencies is used. FIG. 10 shows the energy level of Pr ³⁺ ions of Pr: YSO and the frequency setting of the laser.

First, the beam is split into two by a beam splitter 501, one of which is passed through an acousto-optic effect element (AOM) 102 and irradiated to a crystal in a cryostat 801, which is ± 5 / 2− ± of Pr ³⁺ ions. Resonates with the 5/2 transition and acts as a re-pump (see Figure 10 and the above paper).

  The remaining one laser beam is used to generate two pulses for STIRAP. In the above paper, this was divided into two, and two pulses were obtained by shaping the intensity into Gaussian pulses independently by an acousto-optic effect element (AOM). However, this cuts off some of the laser power. On the other hand, in the pulse waveform of the present embodiment, the sum of the powers of the two pulses is always almost constant, so it is desirable to perform pulse shaping at the stage of division. By doing so, the laser power can be maximized.

  Therefore, the polarized light is rotated through the electro-optic effect element (EOM) 301 and divided into two by the polarization beam splitter (PBS) 401. In this way, the power can be divided at an arbitrary ratio at any time, and pulse shaping and laser division can be performed simultaneously. The basic structure of the pulse of the invention is thus obtained. In order to realize a delicate waveform at the end, two pulses divided by a polarization beam splitter (PBS) 401 are passed through AOMs 101 and 103, respectively. These AOMs are also used to set the frequency difference as shown in FIG. The AOM 102 is used for waveform shaping and frequency setting of the re-pump light.

  In order to execute the pulse shaping and other controls of the present invention using the above EOM and AOM, the controller 1101 operates the EOM and AOM.

  The two pulses thus obtained are irradiated onto Pr: YSO cooled to liquid helium temperature in a cryostat, and the STIRAP of the present invention is executed.

101-103 ... Acousto-optic effect element (AOM)
201-205 ... High reflection mirror 301 ... Electro-optic effect element (EOM)
401... Polarizing beam splitter (PBS)
501 ... Beam splitters 601 to 603 ... Half wave plate 701 ... CW ring dye laser 801 ... Cryostat 901 ... Pr ³⁺ : Y ₂ SiO ₅ crystal 1001, 1002 ... Photodetector 1101... Control device

Claims (5)

For a physical system having the lower two states | 0>, | 1>, and the upper one state | 2>, two pulses that resonate with the | 0>-| 2> transition and | 1>-| 2> transition (hereinafter, A quantum information processing method for changing the state of the physical system from | 0> to | 1> during a movement time T by irradiating pulse 0 and pulse 1), respectively.

The quantum information processing method characterized by satisfy | filling.

The quantum information processing method according to claim 1, wherein:

The quantum information processing method according to claim 1 or 2.

The quantum information processing method according to claim 1, wherein the quantum information processing method is a function obtained by adding a constant to an exponential function at time t. In a quantum information processing apparatus comprising: means for generating a plurality of pulse waveforms for STIRAP comprising an electro-optic effect element, a polarization beam splitter, and an acousto-optic effect element; and a target irradiated with the plurality of generated pulses.
A quantum information processing apparatus that controls generation of the plurality of pulse waveforms so as to satisfy the following condition.

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