Nicolas Aubert

Understanding Employer's Stock Holdings in the French Company Savings Plans Using the Literature on the American 401(k) plans

La faillite de firmes américaines tristement célèbres dont les plans de retraite étaient largement investis en actions de l"entreprise a suscité un intérêt croissant pour la compréhension du choix des salariés d"investir dans des actions de leur entreprise.

Les plans de retraites américains 401(k) ainsi que les Plans d"Epargne Entreprise français ont la caractéristique commune de proposer aux salariés plusieurs catégories de fonds d"investissement. Certains de ces fonds sont investis dans des actions de l"entreprise.

L"objectif de cet article est de mieux comprendre les raisons qui conduisent les salariés à investir dans des actions de leur entreprise alors qu"un tel comportement semble violer les principales recommandations de la théorie moderne du portefeuille.

Afin de mieux comprendre ces choix, nous analysons la littérature américaine qui a mis en évidence leurs déterminants dans le cadre des plans de retraite 401(k). Ces recherches concluent que les choix des employés sont influencés par des déterminants rationnels et comportementaux : les abondements de l"entreprise dans les fonds d"actionnariat salarié, l"extrapolation excessive des performances passées de l"action de l"entreprise, la familiarité et la loyauté des salariés, des heuristiques de décisions et les caractéristiques des salariés et de leur entreprise.

Introduction

Financial participation in France is rooted in the idea of cooperation between capital and labour introduced by De Gaulle. Since the 1950s, French legislation has continuously developed financial participation in several ways. Gainsharing (intéressement des salariés) was introduced in 1959, the compulsory deferred profit-sharing scheme (participation aux bénéfices)1 and the company savings plans (Plan d'Epargne Entreprise) in 1967. In France, employee ownership could hardly be considered separately from the company savings plan.

Most of the employee ownership assets are actually held in these plans. These plans offer the employees the choice to invest between diversified funds and employee ownership funds.

Public data shows that a major part of the company savings plans assets" consist of company stocks suggesting employees are willing to select them.

An important characteristic of employee ownership in France is that it is mainly spread among the large companies quoted in the CAC 40 index 2 . Like other indexes between 2000 and 2002, the CAC 40 experienced a massive decrease. According to a study in "Le Monde" 3 newspaper, the average employee-owner"s portfolio experienced a proportional decrease being divided by two during the same period. Such an event highlighted the underdiversification problem of employee ownership. Indeed, investing massively in one"s company stocks is in contradiction with the basic rules of diversification derived from the modern portfolio theory. Such an investment decision is especially questionable when it is aimed at funding retirement pensions.

The American most popular company based retirement plan, the 401(k) Defined Contribution plan, allows employees to invest in their company stock. According to [START_REF] Mitchell | Company Stock and Retirement Plan Diversification[END_REF] calculations, 11 million plan participants have allocations above 20 percent of their account balance in their company stock and 5 million have allocations above 60 percent of their account balance. A third of the assets in large 401(k) plans are invested in company stock [START_REF] Benartzi | Excessive extrapolation and the allocation of 401(k) accounts to company stock[END_REF]. Using the capital asset pricing model, Meulbroek (2002) shows that diversification cost of company stock is about 42 percent of its value. Option pricing techniques allowed [START_REF] Ramaswamy | Corporate Stock and Pension Plan Diversification[END_REF] to find that the cost of insuring the extra risk of company stock is prohibitively expensive. This under-diversification phenomenon recently aroused the academic interest. These researches are of great interest for a better understanding of the French employees" behaviour regarding their company savings plan. It is worth noting that both the 401(k) plan and the French company savings plan are based upon the same principle, they are participantmanaged. That is, participants are free to choose the way their savings are invested. As stated by [START_REF] Mitchell | Lessons from behavioral finance for retirement plan design[END_REF] such a principle is rooted in an implicit assumption about behaviour: "That the employee-citizen to whom the responsibility of choice has been handed is a well-informed economic agent who acts rationally to maximise its self-interest". It assumes people are able to make the best out of the information available to them. However, employees" decisions about their savings tend to challenge this underlying assumption.

Empirical research about 401(k) plans shows that behavioural determinants also seem to be involved in the employees" decisions.

To comprehend employees" investment decisions about company stock, we will review the American literature, which emphasised their determinants in the context of the 401(k) retirement plans. This literature concludes that employees" choices are led by several rational and behavioural determinants.

As the legal context can strongly influence employees investment decisions about their company stock, we briefly present the specificities of the French Company Savings Plan and underline the main differences with the American 401(k) plan.

Secondly, employers" matching contributions in company stock seem to be a key factor of employees" choice. They are a powerful incentive to buy company stock. Still, according to an "endorsement effect", employees tend to interpret them as implicit advice from their employer to buy company stock.

Thirdly, according to the "excessive extrapolation effect", when past returns on company stock are good enough, employees are more likely to choose company stock and conversely.

Fourthly, employees tend to invest more in company stock when they feel more familiar with their company or more loyal to it.

In section five, we present how plan design can influence employees leading them to use diversification heuristics to make their choice among the investment options available.

Finally, employee"s and company"s characteristics must be taken into account since they can greatly affect investments in company stock.

The specificities of the French Company Savings Plans

French workers have different ways to buy stocks of the company for which they work.

First, employees can invest in their company stocks simply by buying them directly from the financial market. This direct ownership does not give the same privileges of holding shares in a PEE 4 (Plan Epargne Entreprise). Therefore the most common way to become an employeeowner is to invest in a PEE. This company savings plan gives several investment options to the employees. The first available choice is to invest in a diversified fund (FCPE dédié).

Another possibility is to choose a fund invested primarily in company stock. Amounts that can be invested in these two different kinds of funds are the profit-sharing bonuses (both cashbased and compulsory deferred profit-sharing) or the voluntary contributions. Employers can match cash based bonuses and employees" voluntary investments (abondement). The annual amount invested by the employees in the PEE cannot exceed the quarter of their annual salary. According to a recent survey, 90% of the surveyed employees became owners via a PEE5 . As mentioned earlier, a French employee has to choose between two different kinds of funds (FCPE). The first type is diversified funds (FCPE multi-entreprises) whose assets cannot be invested for more than 30% in company stock. The second one is considered as employee ownership since it is mainly invested in the company"s equities (FCPE d'actionnariat salarié ). More than 30% of its assets are invested in the company stock.

French law requires that a PEE should propose at least one diversified fund to the employees in addition to the employee ownership one. (2001), French legislation offers a legal framework and generous tax advantages to a variety of financial participation forms. Concerning the amounts allocated to a company savings plan, if they are frozen for a minimum of 5 years, they are for the most part exempt from taxes. The employers" contributions to the plan are exempt from corporation or income taxes, taxes on wages and social charges, within a yearly limit per employee. For the employees, bonuses allocated to the company savings plan are exempt from income tax.

From a macroeconomic point of view, Poutsma (2001) states that the relationship between financial participation schemes and pension funding has not been investigated in France. The 2003 retirement law introduced a long-run retirement plan and prohibits it to be invested in company stock. From the point of our paper, a major difference between the PEE and the 401(k) plan is that the latter is aimed at funding American workers" retirement pensions. So far, the most popular company based savings plan in France remains the PEE.

But, even if it is not aimed at funding retirement pensions, a recent survey6 shows that 67% of the surveyed employee owners invested their company savings plan in company stock in order to fund their retirement pensions.

Another important difference between the two schemes we are interested in is that the 401(k) does not offer discounts on company stocks. In the context of an employee stock offering, the French Law does allow employers to gratify their employees with discounts on their stock price up to 20% worth. When setting up such financial operations, most French companies attribute the maximum discount combining them with matching contribution in company stock.

The two differences mentioned above are to be taken into account by the forthcoming empirical studies on employees investment behaviour.

In the next sections, we study the literature about company stock holdings in the 401(k) plans.

Matching contribution in company stock

Company"s matching contributions in their stocks appears to be a powerful incentive for the employees. In 2002, among the 401(k) plans in which employers were offering matching contributions in company stock, an average 53% of the total assets were invested in company stock. This percentage drops to 29% within the 401(k) plans offering company stock whatever the employer"s matching contribution policy [START_REF] Holden | 401(k) Plan Asset Allocation, Account Balances, and Loan Activity in 2000[END_REF]VanDerhei (2002)).

Both the French and the American law make these contributions very attractive regardless of the kind of fund they are directed to. But the literature about employee ownership suggests that matching in company stock"s benefits could be more than fiscal for the companies.

Incentive effects implied by employee ownership can result in an increased

productivity. An important literature deals with these incentives effects. These researches assessed how employee ownership affects behavioural dependent variables such as motivation, implication, involvement, satisfaction, turnover and turnover intention. [START_REF] Klein | Employee stock ownership and employee attitudes: a test of three models[END_REF]) identified three perspectives to explain the effects of employee stock ownership on employees" behaviour. The first is the intrinsic satisfaction model. It states that employee ownership per se can increase employees" commitment to the organisation and its satisfaction.

The second model mentioned by [START_REF] Klein | Employee stock ownership and employee attitudes: a test of three models[END_REF]) is the instrumental satisfaction model. It postulates that the employee owner satisfaction and commitment come from the participation in decision-making that is implied by employee stock ownership. Finally, the main hypothesis of the extrinsic satisfaction model is that employee stock ownership is motivating when it is financially rewarding (French (1987)). This latter model is consistent with the idea according to which employees would act rationally. Authors who tested the hypotheses of these three models have showed that the instrumental and the extrinsic models are the most meaningful.

Examining the main research results about the combined effect of financial rewards and employee participation, Blasi, et al. (2003) conclude that the two elements should be paired to improve company"s productivity. In the literature, performance is measured in terms of productivity and profitability. [START_REF] Kruse | Research evidence on the prevalence and effects of employee ownership[END_REF] states that empirical literature considers employee stock ownership as having either a positive or a null effect on performance. Indeed, few studies reported negative effect of employee stock ownership on corporate performance, which contradicts the free rider effect of collective incentives. According to the results of the "Actionnariat salarié 2002" survey, the main motivations that lead employee to invest in employee ownership are the financial contributions and the discounted stocks offered by the company. Another result is that the perception of a high expected stock value is another important motivation of the investment in company stock.

Academic researchers and financial practitioners are well aware that employee stock ownership is a powerful anti-takeover tool since it can reduce the probability of a takeover [START_REF] Shivdasani | Board composition, ownership structure, and hostile takeovers[END_REF]; [START_REF] Beatty | The cash flow and informational effects of employee stock ownership plans[END_REF]). It is considered as a way to put a greater part of the voting rights in the hands of the management. The argument here is that collusion between management and employee owners could happen leading to management entrenchment.

When an employee stock ownership plan is implemented, event studies report negative reaction of the financial market in line with the management entrenchment hypothesis [START_REF] Chang | Employee stock ownership plans and shareholder wealth: an empirical investigation[END_REF]; [START_REF] Chang | Managerial vote ownership and shareholder wealth: evidence from employee stock ownership plans[END_REF]; [START_REF] Conte | Financial returns of public ESOP companies: investor effects vs. manager effects[END_REF]). As a takeover defence mechanism employee stock ownership could be more powerful than poison pill or golden parachutes [START_REF] Chaplinsky | The role of ESOPs in takeover contests[END_REF]). The latter are used less when an employee stock ownership plan is implemented [START_REF] Park | Employee stock ownership plans, firm performance, and monitoring by outside blockholders[END_REF]). Market reactions also depend on other variables such as the board composition and the capital concentration. But the presence of employee stock ownership could also be interpreted as a way to increase the premium obtained by the shareholders of the target company [START_REF] Stulz | Managerial control of voting rights: Financing policies and the market for corporate control[END_REF]; [START_REF] Dhillon | Employee stock ownership and corporate control: an empirical study[END_REF]; Chaplinsky

and Niehaus (1994)). [START_REF] Rauh | Own Company Stock in Defined Contribution Pension Plans: A Takeover Defense?[END_REF] finds additional support to these findings in the more specific context of company stock investment in the 401(k) plans.

Even if a financial incentive favouring employee ownership appears to be costly for the firm in the first place, firms can also benefit greatly from offering them to the employees.

Accordingly, employees" responses to such contributions could be to increase their investment in company stock. With regard to the participants" contributions to every sort of funds available in 401(k) plansand not only to company stock -, [START_REF] Madrian | The Power of Suggestion: Inertia in 401(k) Participation and Savings Behavior[END_REF] found that they are significantly increased when the company offers matching contributions.

Their results suggest that participant"s contributions are increased once the minimal requested tenure to be eligible for matching contributions is reached. Hence, employees wait to be eligible to company"s contribution to start investing in their 401(k). This conclusion applies whatever the kind of funds the matching contribution is directed to. But how employees react when matching contributions are offered in company stock? [START_REF] Benartzi | Excessive extrapolation and the allocation of 401(k) accounts to company stock[END_REF] was the first to investigate employees" investment decision about company stock in the context of the 401(k) plans. He found that matching contribution in company stock could lead to an "endorsement effect" consisting in employees tending to interpret their employer"s contributions as implicit investment advice. [START_REF] Benartzi | Excessive extrapolation and the allocation of 401(k) accounts to company stock[END_REF] findings are confirmed by Purcell ( 2003)"s ones. According to his research, 401(k) assets concentration in company stock is significantly correlated with matching contribution in company stock practices. Liang and Weisbenner (2002) findings are consistent with

Benartzi"s ones. According to them, employees put a larger share of their own contributions in company stock when the company"s matches are offered in company stock. Moreover, a switch from allowing the employee to invest in the match, without restriction to requiring that the match be all in company stock, is not offset by the employee investing less of his own contributions in company stock.

In France, most of the advantages offered to the employees within the context of employee stock offerings are given through the company"s saving plans. For instance, employees could benefit from discounted company stock if they buy company stock through their company"s saving plan. This is one of the major differences with the 401(k) plan. Actually, employees cannot be offered company stock at a discounted price. Discount offered on company stock price may also be considered as a powerful incentive to buy company stock. Some French listed companies can be extremely generous offering matching contributions in company stock. In the context of an employee offering, the French Law allow employers to gratify their employees by discounting their stock price up to 20% worth. When such financial operations happen, most companies attribute the maximum discount to their employees. According to the French Market Authority (AMF), 52 listed companies organised employee stock offerings in 2001, 52 in 2002, 47 in 2003 and 48 in 2004.

Excessive extrapolation of past returns

Benartzi (2001) identified employees" decisions about their company stock with the "representativeness" bias highlighted by [START_REF] Tversky | Judgment under uncertainty: Heuristics and biases[END_REF]. According to representativeness, [START_REF] Benartzi | Excessive extrapolation and the allocation of 401(k) accounts to company stock[END_REF] argues that employees tend to conclude that abnormally high past performance is representative of future performance, even if stock returns are largely unpredictable. Such a cognitive bias is consistent with the existence of an "excessive extrapolation" phenomenon. According to excessive extrapolation, employees whose firms experienced a good (bad) stock performance are more likely to invest a greater (lower) fraction of their 401(k) plan in company stock. His empirical results are consistent with an excessive extrapolation effect and are highly significant for a preceding period of ten years.

For this latter period, [START_REF] Benartzi | Excessive extrapolation and the allocation of 401(k) accounts to company stock[END_REF] shows that employees whose firms experienced the best returns during the last ten years invested nearly 40% of their plans in company stock.

Conversely, for the same period of time, employees whose firms experienced the worst results invested only 10% of their plan in company stock. Finally, he tested his excessive extrapolation hypothesis using a survey. The latter confirmed his result findings that employees consider past returns on company stock to be correlated with future returns.

Investigating 401(k) plans participants" new inflows and transfers, Huberman and Sengmuller (2004) found they can be predicted by company stock"s past returns over a three-year period.

Nevertheless, their results suggest that sensibility to past returns is asymmetrical. Actually, investors would react more strongly to positive and above S&P 500 returns than to negative returns. Surprisingly, both [START_REF] Benartzi | Excessive extrapolation and the allocation of 401(k) accounts to company stock[END_REF] and Huberman and Sengmuller (2004) found that employees" decisions are not affected by the standard deviation of their company stock. [START_REF] Choi | Employees' Investment Decisions About Company Stock[END_REF] used data about 401(k) plans participants within three large companies. Their findings are consistent with an excessive extrapolation effect for only one-year past returns period. Past returns appear to matter at every stage of employees" decisions: initial contributions, subsequent changes in contributions fraction and trading behaviour. They report that high returns lead participants to invest a greater part of their portfolioboth initial contributions and subsequent fractions of new inflows -in company stock. However, high returns have a converse effect on their trading behaviour. Actually, they induce employees to reallocate their portfolios away from company stock and toward other equities. According to them, 401(k) participants would follow the trend when making decisions about investment flows whereas they would be "contrarian" investors regarding their trading behaviour. This finding is consistent with a "profit-taking" behaviour.

Employees' loyalty and familiarity

Despite the obvious diversification gains investment in foreign securities could bring about, studies have shown that investors are reluctant to include them in their portfolio. The common explanation is that investors prefer securities with which they are more familiar. This familiarity bias was tested by [START_REF] Huberman | Familiarity breeds investment[END_REF]. Even if his research did not investigate employees" decisions about company stock, he considered this diversification problem as potentially resulting from a familiarity bias. According to [START_REF] Driscoll | Employee perspectives on stock ownership: financial investment or mechanism of control?[END_REF] findings, employees consider investing in their company stock to be less risky than investing in a diversified portfolio of American stocks or international stocks. [START_REF] Benartzi | Excessive extrapolation and the allocation of 401(k) accounts to company stock[END_REF] tested interaction effect between familiarity and excessive extrapolation. He finds that employees who feel more familiar with their company are more likely to extrapolate past returns excessively. Consistently with Driscoll, J., [START_REF] Driscoll | Employee perspectives on stock ownership: financial investment or mechanism of control?[END_REF] results, he also shows that employees consider their employer"s stock to be less likely to lose half of its value than overall market portfolio. Yet, Both Huberman and Sengmuller (2004) and [START_REF] Benartzi | Excessive extrapolation and the allocation of 401(k) accounts to company stock[END_REF] state that employees" decisions about company stock cannot be explained by their private information on its future returns. Their findings suggest that employees" decisions do not reflect an ability to predict company stock returns.

Cohen ( 2004) tested the effect of employees" loyalty on their decision about company stock. She assumed a more loyal employee would be more willing to buy her company stock.

Using divisional employee status (conglomerate employee or stand-alone employee) as a loyalty proxy, she finds support of a loyalty based explanation of employees" holdings in company stock. In her study, stand alone employees are assumed to be more loyal than conglomerate ones. She also takes into account other variables such as excessive extrapolation, information based explanation and traditional risk diversification framework.

Using items from the [START_REF] Mowday | The measurement of organizational commitment[END_REF] organisational commitment questionnaire as loyalty proxies, [START_REF] Benartzi | Excessive extrapolation and the allocation of 401(k) accounts to company stock[END_REF] found it has no effect on employees" decision to buy company stock.

As the French have limited experience with direct equity holding, they more likely tend to be subject to familiarity and loyalty bias and by investing in their employer stock. The French government initiated this trend. Indeed during the wave of privatizations in 1986 using employee ownership as a way to develop individual equity holding. Furthermore, the French Law mentions that a part of every public offering could be reserved for the employees.

Plan design effect

Several researches about discretionary 401(k) plans have put the emphasis on the fact that employees" choices are strongly influenced by a "framing effects" [START_REF] Mitchell | Lessons from behavioral finance for retirement plan design[END_REF]). Thus, employees could select different investment options if the plan"s menu design is modified. Such a phenomenon led [START_REF] Benartzi | How Much Is Investor Autonomy Worth?[END_REF] to ask the following question: how much is the freedom of choice given to 401(k) participants worth? They conducted an experience in which plan participants were given the choice between holding their own portfolio, the portfolio of the median participant and the portfolio of the average participant. Surprisingly, eighty percent of the participants preferred the median portfolio to the one they constructed themselves. They conclude that participants do not really know their preferences and do not assess the mean-variance of their portfolio very well. This leads them to select sub-optimal asset allocation with regard to the modern portfolio theory. Benartzi &

Thaler explore an explanation inspired by the "extremeness aversion" first emphasised by [START_REF] Simonson | Choice in context: Tradeoff contrast and extremeness aversion[END_REF]. They define this heuristic as the "tendency for consumers to prefer an option that does not appear to be at the extreme point of some relevant continuum" (p. 1607). This heuristic could be illustrated by the tendency to avoid the most expensive and the least expensive option in a menu. The results of their experience confirm the hypothesis according to which participants choose their portfolio allocation avoiding extreme options i.e.

the most risky and the least risky ones.

In another research, [START_REF] Benartzi | Naive diversification strategies in retirement saving plans[END_REF] that these choices are selected regardless to the funds" composition. For people given the choice between an equity fund and a bond fund, the average allocation to equities was 54 % and so on. According to these findings the design can strongly affect the composition of the participants" portfolio. Liang and Weisbenner ( 2002) investigated 1/N heuristic and employees" decision about company stock. They found that naïve diversification could apply and can predict employees" allocation to company stock. They also emphasised that employees adapt their naïve diversification behaviour in response to an increase of the number of funds available five years after such an increase.

Further findings argue for a strong effect of the numbers of funds offered in the plans.

According to [START_REF] Iyengar | How much choice is too much? Contributions to 401(k) retirement plans[END_REF], adding investment options to a plan would decrease participants" contributions to it. This "choice overload" would result in a 1,5% to 2% drop in participation rate for every ten funds added.

Other [START_REF] Benartzi | Risk aversion or myopia? choices in repeated gambles and retirement investments[END_REF] experiences results are consistent with [START_REF] Kahneman | Choices, Values and Frames[END_REF] conclusions according to which the way information is displayed can affect individuals" choices. According to their experiments, investors react differently according to whether the long-run results or short-run results are presented.

In France, the PEE must at least propose one fund in addition to employee ownership.

Before the introduction of this reform, many French companies gave no choice but employee ownership to their employees. As a result, profit sharing bonuses could only be invested in company stock.

As mentioned earlier, there are two main differences between the 401(k) and the PEE.

First, the PEE is not aimed at funding retirement pensions. Moreover, employees can get discounts on company stock price in the context of an employee stock offering. From the investor point of view, investing five years or forty years is totally different. In the United

States, a consequence of the longer freezing period is that investors are given a larger choice.

According to [START_REF] Iyengar | How much choice is too much? Contributions to 401(k) retirement plans[END_REF] some American employees could select among more than fifty different funds. None of the PEEs gives such a large choice to French employees.

Another important difference between the French and the American schemes is that the former allows employee stock offering. Some of the largest French listed companies are offering shares to their employees on a regular basis, sometimes several times a year. During these operations, French firms can favour different goals. For instance, they can favour the participation rate by offering leveraged employee ownership funds. Consequently, employees can greatly increase their investment capacity by investing in these funds. They allow lower paid workers to participate in the employee stock offering more easily. These leveraged employee ownership funds are all the more attractive that they are sometimes downside protected. Another objective could be to maximise the total amount invested by the employees. To do so, French firms can schedule employee stock offering just after the payment of the profit sharing bonuses. Indeed, although compulsory profit sharing bonuses are not entitled to matching contribution, in the context of an employee stock offering employees can benefit from a discount. Because of these French specificities, employee ownership investment is highly attractive.

Employees' and companies characteristics

Both employees" and companies" characteristics can affect employees" choices regarding their investment in company stock. For instance, from the modern portfolio theory point, individuals" portfolio could vary with the employees" age and risk aversion. On another hand, company stock"s sensibility to the market index should be taken into account for a better understanding of employees" choices. At the pension funds level, Even and Macpherson (2003) results suggest that investment behaviour is consistent with predictions generated by models of optimal portfolio management. Investment in company stock is avoided when the non-diversification costs (assessed with respect to the Capital Asset Pricing Model) are too high. Accordingly, we could observe that American "blue chips"" 401(k) plans assets are massively invested in company stock. They are usually supposed to be less risky than smaller firms.

Following the portfolio theory"s arguments, it is worth noting that, by investing in company stock, employees are considering not only their financial wealth but rather their overall wealth such as their estate property, their salary or their human capital. Poterba (2003) model suggests that employees" risk aversion and all the components of their wealth should be taken into account to better understand their investment choices. Mitchell and Stephen P.

Utkus (2002) argue that investing in company stock could be considered as rational if a company offers other retirement plans, such as Defined Benefit plans, to its workers. From this point, only the money invested in the 401(k) is at risk whereas the Defined Benefit plan is not. According to them, out of the 96 largest 401(k) plans, all but one also offers such a plan.

Among the assets invested by employees in their company, the most important one is their human capital. The major risk associated with employee ownership is to lose both one"s job and savings. Specificity of human capital invested by an employee in his firm could play a role in his decision to buy company stock. The more specific the human capital is, the less likely is the employee to buy company stock. Another variable to consider is employees" wage. [START_REF] Huberman | Defined contribution pension plans: determinants of participation and contributions rates[END_REF][START_REF] Agnew | Portfolio choice and trading in a large 401(k) plan[END_REF] state that higher salary bring about higher contributions in the 401(k) plans. In the context of the France Telecom"s privatization, [START_REF] Degeorge | Selling Shares to Reluctant employees: France Telecom's Experience[END_REF] found that employees with higher wages had participated more in the employee"s offering. As the employees" specific human capital proxies they used the employee"s status (civil servants vs. non civil servants, former employees vs. current employees, retirees vs. current employees) and job tenure. As far as the specificity of human capital is concerned, their results suggest employees" status did not matter in the employees" decision to participate in the offering. Regarding job tenure, Degeorge et al. found it was significantly related to the decision to buy company stock. They also measured how a proxy of employees" wealth (zip code) could affect employees" decision to participate and how much to invest in France Telecom stock. This variable appeared to be significantly correlated with the decision to buy company stock. Thus employees with higher wealth would have been more likely to buy company stock. To our knowledge, [START_REF] Degeorge | Selling Shares to Reluctant employees: France Telecom's Experience[END_REF] is the only study aimed at understanding employee"s decision about their company stock in a French context. [START_REF] Barber | Boys will be boys:gender, overconfidence, and common stock investment[END_REF] findings show that investment behaviour could differ according to the gender. Huberman, [START_REF] Huberman | Defined contribution pension plans: determinants of participation and contributions rates[END_REF] and [START_REF] Agnew | Portfolio choice and trading in a large 401(k) plan[END_REF] also found a gender effect in the employees" decisions about their 401(k) plans. Women participation"s rate tends to be higher than men"s one, they invest more in liquidity than stocks and the number of their transactions is lower. Another variable closely related to gender is marital status. Indeed, one can argue that the risk to invest in company stock is decreased if an employee"s partner gets revenue elsewhere. This could result in a diversification effect. [START_REF] Degeorge | Selling Shares to Reluctant employees: France Telecom's Experience[END_REF] report an effect of the gender on the probability to participate. Women participated more than men in the France Telecom employee offering. The authors mention that marital status caused this difference. [START_REF] Degeorge | Selling Shares to Reluctant employees: France Telecom's Experience[END_REF] put the emphasis on the France Telecom"s marketing effort before the initial public offering. As for them, a key part of this marketing campaign was aimed at the employees. Many different benefits were offered to the employees: free or discounted stocks, low-rate loans. This underlines the importance of the company"s communication policy toward its employees. It can be assumed that such communication policies provide a good incentive for employees to buy company stock. As mentioned earlier, the French Law requires that at least one investment option is given to employees in addition to employee ownership. This rule was aimed at giving diversification opportunities to employees but some French companies are still focusing their communication policy on employee ownership. Such a behaviour strongly affect employees" choices regarding their investment. More specifically, during an employee stock offering, a firm can use several means of communication. For instance, internal communication can be done via intranet, posters, leaflets or information meetings. Communication campaigns are often outsourced to

HRM consultants who answer employees" questions directly in the workplace or via a dedicated hotline. Sometimes, they even help employees to fill up their subscription form.

A similar argument about companies" marketing effort would be that communication and advertising campaign aimed not only at the employees but at company"s customers could also affect employees" decision about company stock. Assuming that such advertising expenses make employees proud of their company, [START_REF] Cohen | Loyalty based portfolio choice[END_REF] reported that these expenses are positively correlated with employees" holdings in company stock.

Conclusion

As participant-directed defined contribution plans are becoming more and more adopted in the private-sector retirement system throughout the world, understanding what are the determinants of participants" investment decision is of great interest. Researches dealing with this topic involve several disciplines of management and, more broadly, of human sciences. In particular, the empirical literature about employees" decisions regarding their employer stock suggests that forthcoming empirical studies should supplement the rational assumptions with behavioural variables. More broadly, investors" behaviour is a good ground to assess their explanatory power empirically. Such researches could lead to enhance plans" design in order to increase participants" welfare. A good example of that kind of research is Benartzi & Thaler (2004). Taking into account empirical conclusions about 401(k)

participants" savings behaviour, they designed a new sort of plan: "Save More Tomorrow".

Improving plan design in the best interest of the participants implies acknowledging that they are barely able to make decisions consistent with modern portfolio theory. This is consistent with Sunstein & Thaler"s ( 2003) "libertarian paternalism" they define as follows:

"Libertarian paternalism is a philosophy that advocates designing institutions that help people make better decisions but do not impinge on their freedom to choose."

Regarding the employees" stock holdings in company stock, it seems paradoxical to notice that they have greatly contributed to the success of the 401(k) plans and the French Company Savings plans so far. Employers can take advantage of employee ownership, which induces them to contribute to the plans. Recently, the French government introduced a new company based retirement plan: the PERCO (Plan d"Epargne de Retraite Collectif). This plan replaced the PPESV (Plan Partenarial d"Epargne Salariale Volontaire) prohibiting investments in company stock. This restriction will solve the diversification problem but employers will probably be less willing to contribute to these plans than to the Company Savings Plans or to the former PPESV. Eventually, one should consider that the cause of the 401(k) plans and the French CSP drawbacks is also the cause of their success. Although it implies increased risk for the employees, the possibility of giving matching contributions in their own stock is a powerful way to get companies involved in the funding of retirement pensions. At the company level, employee ownership is also likely to have effects on employees" attitudes at work and corporate governance. But this effects are both closely related to returns on company stock.

Taking into account the American conclusions is of great interest to better understand French workers investment decisions. To do so, the institutional specificities should not be underestimated. Particularly, the French Company Savings Plans are not aimed at funding retirement pensions and they are designed to allow employee stock offerings. In spite of these differences, the literature presented above could be considered as a theoretical framework by the forthcoming empirical researches.

  underlined another diversification heuristic: "Naïve diversification or 1/N heuristic". Applying naïve diversification consists in allocating an equal fraction 1/N of a new contribution in each fund available in the plan. Their experiments show that, when participants are proposed two funds -whatever the securities" mix: bonds & stocks, stocks & balanced funds, bonds & balanced fundstheir common investment strategy is to choose a 50/50 mix of the two funds offered. It is striking to observe

Table 1 :

 1 Table 1 shows the composition of all the FCPEs" assets in 2003. These assets are mostly invested in company stock. FCPE assets, Source: Rapport AMF 2004.

		Euros	Percent
		(billions)
		22,5
	Quoted Employer's stock
	Unquoted Employer"s stock	2,5
	Employer"s bond	2
	Other stocks	8,7
	Liquidity	6,8
	Other Mutual Funds	14,4
	TOTAL	56,92	100%
	Investing in a PEE is financially rewarding from the employee"s point for at least two
	reasons. The first is that money placed in the PEE by the employee can be matched by an
	employer"s contribution as mentioned earlier. By adding a contribution, a company can
	favour investment in one or several available funds. Furthermore, in the case of an employee
	stock offering, employees who buy stocks could take advantage of a maximum discount of
	20% on the stock"s price. There are therefore two different types of financial contribution that
	can give the employees an incentive to buy company stock. The counterpart is that money

This compulsory profit-sharing scheme is a characteristic feature of the French participation system. All companies with a minimum workforce of over 100 (and since 1994 of over 50) are required to institute a deferred profit-sharing fund. The compulsory profit sharing bonuses are frozen five years.

In 2003, the French Financial Market Authority (AMF) report mentioned that 40% of the FCPE assets are invested in the largest French companies.

Le Monde,

octobre 2002.

The decision of implementing a PEE is made only by the manager.

Enquete « Actionnariat Salarié 2002 » realized by Hewitt Associates in partnership with Société Générale Asset Management.

Enquête "Actionnariat Salarié 2002", Hewitt Associates et Société Générale Epargne Entreprise.


Ivan I Maximov

Niels Chr Nielsen

Julien Salomon

Gabriel Turinici
email: gabriel.turinici@dauphine.fr

A smoothing monotonic convergent optimal control algorithm for NMR pulse sequence design

The past decade has demonstrated increasing interests in using optimal control based methods within coherent quantum controllable systems. The versatility of such methods has been demonstrated with particular elegance within nuclear magnetic resonance (NMR) where natural separation between coherent and dissipative spin dynamics processes has enabled coherent quantum control over long periods of time to shape the experiment to almost ideal adoption to the spin system and external manipulations. This has led to new design principles as well as powerful new experimental methods within magnetic resonance imaging, liquid-state and solid-state NMR spectroscopy. For this development to continue and expand, it is crucially important to constantly improve the underlying numerical algorithms to provide numerical solutions which are optimally compatible with implementation on current instrumentation and at same time are numerically stable and offer fast monotonic convergence towards the target. Addressing such aims, we here present a smoothing monotonically convergent algorithm for pulse sequence design in magnetic resonance which with improved optimization stability lead to smooth pulse sequence easier to implement experimentally and potentially understand within the analytical framework of modern NMR spectroscopy.

I. INTRODUCTION

Optimal control theory (OCT) is a powerful method for control and design of processes within quantum dynamics.

Originally the method was applied for problems within engineering and economics. [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF][START_REF] Krotov | Global methods in optimal control theory[END_REF] During the past decade or so, optimal-control-based methods have been increasingly used for a development of new experiments within optical spectroscopy, [START_REF] Tannor | Time-Dependent Quantum Molecular Dynamics[END_REF][START_REF] Zhu | [END_REF][5][6][7][8] quantum information processing, [9][10][11][12][13][14] liquid-and solid-state nuclear magnetic resonance (NMR) spectroscopy, [15][16][17][18][19][20][21][22][23][24][25][26][27][START_REF] Nielsen | Encyclopedia of Magnetic Resonance[END_REF][START_REF] Tošner | [END_REF] magnetic resonance imaging (MRI), [30][31][32][START_REF] Anand | Proceedings of Modeling, simulation and optimization of complex processes Conference[END_REF][START_REF] Lee | [END_REF][35][36] and dynamic nuclear polarization (DNP) hybrids between electron and nuclear magnetic resonance. [37][38][39][40] Such applications have not only been useful for the specific disciplines taking advantage of new efficient design procedures and improved experimental methods, but it has also stimulated mathematical investigations in quantum optimal control theory. [41][42][43][44][45][46][START_REF]Introduction to Quantum Control and Dynamics[END_REF] The latter addresses fundamental questions concerned with controllability, convergence, and the establishment of powerful numerical methods for optimal control in quantum systems.

So far, the vast majority of optimal control applications within magnetic resonance have taken advantage of gradientbased methods, such as the gradient ascent pulse engineering (GRAPE) algorithm introduced for NMR applications by Khaneja and coworkers, [19] and recently further developed and distributed for general use by Nielsen and coworkers in an optimal control version [21,[START_REF] Tošner | [END_REF] of the open-source NMR simulation software SIMPSON. [START_REF] Bak | [END_REF]49] In combination with conjugated gradient algorithms, this method [19,21,24,[START_REF] Tošner | [END_REF] proves to be very powerful, as demonstrated by numerous applications in which NMR methods with improved experimental sensitivity, robustness towards variations in instrumental or spin system parameters, reduction of the undesirable effects from dissipative processes (i.e., relaxation), and experiments with lower radio-frequency (rf) power requirements and thereby reduced risks for sample heating have been developed. More recently, it has been demonstrated [38] that a monotonic convergent variant of optimal control method based on the algorithms of Krotov [START_REF] Krotov | Global methods in optimal control theory[END_REF][START_REF] Tannor | Time-Dependent Quantum Molecular Dynamics[END_REF][START_REF] Zhu | [END_REF]6] represents an interesting alternative to the gradient-based approaches for efficient experiment design. Our initial work with this algorithm, in a density operator formulation, exposed important computational properties of optimal control methods such as global extremum searching, fast convergence, algorithmic simplicity, and independence on time discretization in terms of convergence.

Despite increasing use, it is apparent that current methods face serious challenges in the practical realization which needs to be addressed to exploit the full potential of optimal control based methods for design of optimal experiments. This applies not least for the most challenging purposes involving large spin systems, optimizations ensuring broadband or band-selective operation with respect to certain spin system parameters (e.g., chemical shifts), and powder samples in solid-state NMR spectroscopy. For example, looking at the many optimal control pulse sequences proposed so far, it appears that many of these display quite wild oscillations in phase and amplitude of the rf control fields (see, e.g., Refs. [18,[20][21][22][23][24][25][26][27][START_REF] Nielsen | Encyclopedia of Magnetic Resonance[END_REF][START_REF] Tošner | [END_REF]) which may complicate implementation on available instrumentation with limitations on the speed and accuracy of phase and amplitude switching. Furthermore, it turns out that GRAPE displays a quite strong dependence on the initial guess of the pulse sequence, dependence on the applied time discretization (i.e., the number of pulse variables, and their duration), as well as unpredictable convergence to local extremum points. Along the same lines, the monotonic Krotov-based algorithm faces problems with numerical instability and difficulties in a selection of parameters controlling the flow of operations and balance of necessary running costs. [38] These problems have introduced undesired needs for intuition, experience, and repetition of optimizations with a very large set of different initial guesses in the usage of these methods as replacement for stronger mathematical recipes. Although part of these problems have been overpassed in recent adoptions for wave function formalism in optical spectroscopy, [44] a strong need for solutions to the problems still exists for magnetic resonance applications typically performed within a density operator formalism.

In this paper, we present a modified monotonic algorithm which stabilizes convergence and smooths the resulting NMR pulses sequences through the use of a frequency truncation technique in course of the optimization. The latter aspect is practically important realizing that most optimal control sequences presented so far within NMR spectroscopy display quite significant and fast amplitude and phase modulations, which may cause unnecessary problems upon implementation on available spectrometer hardware. This work builds on related techniques introduced in the field of the laser control of alignment and rotation [50] for an unique controlling field.

II. THE OPTIMAL CONTROL PROBLEM

The most typical setup for optimal control pulse sequence design in NMR spectroscopy involves systematic generation of optimal radio-frequency (rf) pulse sequences which in a given spin system either (i) accomplish the most efficient transfer of coherence or polarization from a given initial spin state ρ 0 to a desired target spin state C (often referred to as state-to-state transfer) or (ii) synthesize a specific effective (or average) Hamiltonian [51,52] emphasizing or suppressing certain parts of the internal nuclear spin interactions to tailor the Hamiltonian for evolution under desired interactions. These interactions may provide a desired state-to-state coherence/polarization transfer, specific information, or provide spectral simplification, e.g., in the form of improved spectral resolution.

In absence of dissipative processes, the dynamics of the nuclear spin system may be described by the Liouville-von Neumann equation

dρ(t) dt = -i[H(t), ρ(t)], (1)
where ρ(t) is the density matrix (initial state: ρ(0) = ρ 0 ) and H(t) is a Hamiltonian of the spin system. In the high-field approximation, the latter takes the form

H(t) = H 0 + k ω k (t)H k , (2)
with the first term collecting all internal nuclear spin interactions (chemical shifts as well as J, dipole-dipole, and quadrupole couplings) and the latter describing external rf manipulations with the amplitude ω k (t) (in angular frequency units) for the spin operator H k (typically H k = I x , I y , S x , S y for an I-S two-spin system) being our control fields. The solution to the equation of motion in Eq. ( 1) is typically expressed as

ρ(t) = U (t)ρ 0 U † (t), (3)
where the unitary operator (or propagator) U (t) = D exp -i t 0 H(t ′ )dt ′ links the unitary evolution with the Hamiltonian in Eq. ( 2). D is the so-called Dyson time-ordering operator.

Optimal control relies on optimization of a functional of the type

J i (ω) = φ i - k λ k T 0 ω 2 k (t)dt, (4)
with the first term denoting the final cost (or the objective) and the latter term the running cost considering the collected energy/power used to reach the optimum. The influence of the running cost is scaled by a so-called penalty factor λ k (which may be constant, as assumed here, or time-dependent if, e.g., specific rise and fall time behaviour of the pulses are desired). This convex running cost improves the convergence of the optimization methods. Furthermore, it facilitates development of pulse sequences without too excessive rf power consumption.

For state-to-state transfers between Hermitian operators C and ρ 0 , the final cost (i.e., the overall transfer efficiency) may be expressed as

φ 1 = Tr (Cρ(T )) , ( 5
)
where T is the overall duration of the experiment. For transfers between non-Hermitian operators the final cost may be formulated as [START_REF] Tošner | [END_REF]38]

φ 2 = |Tr C † ρ(T ) | 2 . ( 6
)
For synthesis of a desired propagator U D , the final cost is given by

φ 3 = Re Tr(U (T )U † D ) . (7)
We note that the objectives φ i given above is by no means exclusive, and other variants for the target functions may be used. For example, instead of Eq.( 7) one may use the squared expression

φ ′ 3 = |Tr(U † D U (T )| 2 .
The preferential form of the target function typically depends strongly on the given optimization problem. [15,19,24,[START_REF] Nielsen | Encyclopedia of Magnetic Resonance[END_REF][START_REF] Tošner | [END_REF] III. MONOTONIC CONVERGENCE

The aim of optimal control experiment design is for a given period of time T and a given time discretization (i.e., number of pulses, typically taken equidistantly over the time T ) to find rf control fields (i.e., ω k (t)) which lead to the maximum of the functional in Eq. ( 4). The Krotov-based monotonic algorithms accomplish this using a Lagrange approach with an adjoint propagator B(t), or a Lagrange multiplier for an unconstrained functional as described in Ref. ([38]), using a combination of forward propagation with U (t) (Eq.( 3) ) and backward propagation using a conjugated equation for B(t). We note that different formulations exist for such calculations, including that of Tannor and coworkers [START_REF] Tannor | Time-Dependent Quantum Molecular Dynamics[END_REF] closely following the original formulations of Krotov, a different variant by Zhu et al. [START_REF] Zhu | [END_REF], and a more general approach (embracing the methods of Tannor and Zhu) as described by Maday et al. [6] in context of waveforms and Maximov et al. [38] in context of density operators. For simplicity, we here restrict to the original Krotov approach as formulated by Tannor et al. [START_REF] Tannor | Time-Dependent Quantum Molecular Dynamics[END_REF] (corresponding to δ = 1 and η = 0 in Refs. ( [6]) and ( [38])), while noting that the more general formulation with arbitrary δ and η parameters may readily be implemented at the expense of slightly more complicated formulas.

A prerequisite for monotonic convergence is that the target operator C is positive semi-definite. While this condition is obviously fulfilled for optimization of effective Hamiltonians, it will, for example, for optimization of transfer between Hermitian operators require the final cost function to be modified to φ = φ + κTr U (T )U † (T ) as proposed

previously. [38] The proof for monotonic convergence, in the case of Hermitian operators, may readily be established considering the following decomposition of the variation in the cost functional values between arbitrary rf pulses controls ω and ω ′ (in a discrete representation):

J(ω ′ ) -J(ω) = Tr (U ′ N -U N )(U ′ † N -U † N ) + Tr C(U ′ N -U N )ρ 0 (U ′ † N -U N ) + 2 N +1 j=2 Re Tr e i∆t k ω k,j-1 H k e -i∆t k ω ′ k,j-1 H k -E AU ′ j-1 B j-1 A † -∆tλ k (ω ′ k,j-1 -ω k,j-1 )(ω ′ k,j-1 + ω k,j-1 ), (8)
where we have applied the second-order Strang method to evolve the propagator U j and the Lagrange multiplier B j . [44] The time step ∆t is defined through the number of bins N and the overall time T = N ∆t, while the matrix exponent A is defined as A = e -0.5i∆tH0 . E is the identity operator. We assume all functions to be constant during the time step ∆t rendering computation of the matrix exponents straightforward.

The discrete set of propagators (U j ) and Lagrange multipliers ( B j ) may be obtained using the following equations

U j+1 = Ae -i∆t k ω kj H k AU j , U 0 = E, (9)
B j = B j+1 Ae -i∆t k ω kj H k A, B N = Θ(C), (10)
with the operator Θ(C) defined by Eqs. ( 5)-( 7) and the desired efficiency. For example, for Hermitian state-to-state transfer

Θ(C) = κU N + CU N ρ 0 , ( 11
)
where κ is an insignificant scaling factor. [38] To ensure positiveness of the functional difference J(ω ′ )-J(ω) at each time step, we need to maximize the last string of Eq.( 8). Following the approach previously described for wave functions, [44] it appears convenient to re-express the two last terms of Eq.( 8) as a function f j (ω ′ j ) depending on a vector variable ω ′ j of dimension k relating to the time step j,

f j (ω ′ j ) = 2Re Tr e i∆t k ω k,j-1 H k e -i∆t k ω ′ k,j-1 H k -E AU ′ j-1 B j-1 A † -∆tλ k (ω ′ k,j-1 -ω k,j-1 )(ω ′ k,j-1 + ω k,j-1 ). ( 12
)
In this formulation, the updated control fields ω ′ k,j should be found locally (in time) through a minimization problem of dimension k for the vector function -f j (ω ′ j ) (corresponding to maximization of f j (ω ′ j ) and the corresponding functional in Eq.( 8)), which due to the non-commutative relationships of the operators H k in the matrix exponents can not straightforwardly be simplified further. [6,44] The minimization may be conducted starting with an appropriate initial guess, using routines such as conjugated gradients or quasi-Newton. [START_REF] Press | Numerical Recipes 3rd Edition: The Art of Scientific Computing[END_REF] The iteration equation for the rf controls ω ′ j may be expressed as

ω new j = arg min ω ′ j -f j (ω ′ j ) , (13)
with the rf fields ω j from a previous iteration step used as an initial guess for the minimization problem Eq.( 13).

Equation ( 13) guarantees monotonic convergence of the functional J(ω) for all time steps in each iteration step through solution of an unconstrained nonlinear optimization problem. Indeed,

f j (ω ′ j = ω j ) = 0, so that -f (ω new j ) ≤ 0.
In other words, there exist a pulse sequence ω new j such that J(ω new j ) ≤ J(ω j ). Moreover, note that if the algorithm finds that J(ω new j ) = J(ω j ) then every local optimization procedure has failed to find a strictly better control. In this case, a local maximum has been found. Note also that the previous approach is independent of the optimization method used in the local optimization problems in Eq.( 13).

IV. FREQUENCY CONSTRAINING AND SMOOTHING

Direct application of the monotonic algorithm outlined above, as well as previous gradient-based algorithms, often lead to optimal pulse sequences with significant oscillations in the rf field amplitudes and phases. [18,20,23,25,[START_REF] Tošner | [END_REF]38,40] These variations not only hamper analytical understanding of the function of optimal pulse sequences but may also complicate practical implementation on available instrumentation. To stimulate generation of smoother solutions, we here demonstrate that the optimal control algorithms may be combined with standard frequency truncation techniques with a regularization substep that retains monotonic convergence.

Given ω, an arbitrary control, suppose that a better control ω new has been obtained, i.e. J(ω) < J(ω new ). A way to define a smoother improving control ω smooth is to consider an interpolation between ω new (i.e., those typically displaying significant oscillations) and a regularized version of it (cf. Ref. [50]). In this way define for all k

ω smooth k = (1 -α)ω new k + αF(ω new k ), ( 14
)
where α is a regularization parameter and F(ω) defines the frequency truncation. The frequency truncation may be accomplished in many ways, e.g., using built-in functions in MATLAB [62] or simply by Fourier transforming the pulse sequence, removing high-frequency components, and transforming it back to the time domain using an inverse Fourier transformation.

Following the proofs in the previous section, it is obvious that α = 0 ensure monotonic behaviour and thereby

J(ω smooth α=0 ) ≥ J(ω).
The same is almost true when α → 0 as well. Accordingly, an iterative sub-algorithm may be established which generates a smooth and monotonic solution:

1. start from α = 1;

2. test J(ω smooth ) ≥ J(ω)?

3. if not, decrease α, for example, halving the value and go to step 2.

With this ingredient and the formula in Eqs. ( 9) -( 11), we can formulate the overall algorithm. Consider a parameter T ol > 0.

1. Set the initial random guess ω 0 = (ω 0 j,k ); j = 1, .., N ; k > 0.

2. Compute the corresponding state U 0 and B 0 according to Eqs. ( 9)-(10).

-α ℓ ) ω ℓ+1 + α ℓ F ( ω ℓ+1 )) ≥ J(ω ℓ ) c. Define ω ℓ+1 = (1 -α ℓ ) ω ℓ+1 + α ℓ F ( ω ℓ+1 ) d. Set ǫ = J(ω ℓ+1 ) -J(ω ℓ )
e. Do backward propagation to compute B ℓ+1 the solution of Eq. ( 10) with ω = ω ℓ+1 f. Do ℓ = ℓ + 1

End While.

It is important to note that frequency truncation technique with regularization could equally well be applied to other known optimal control approaches, such as GRAPE and different variants of Krotov-based optimal control.

V. DESIGN OF SMOOTH NMR EXPERIMENTS

While the overall applicability of optimal control for NMR experiment design has been demonstrated and verified numerically and experimentally in numerous previous papers, [15-17, 19-22, 24, 26-29] we will demonstrate numerically in this paper the implication of our smoothing procedure and monotonic convergence in optimal control design of a set of simple NMR experiments. We address specifically spin-state selective, non-Hermitian coherence transfer through J couplings for liquid-state NMR applications and excitation of the central transition of 23 Na for applications in magnetic resonance imaging (MRI). Lists containing rf amplitudes and phases for the optimal control pulse sequences as well as Matlab scripts used to generate these can be found in Supplementary Material. [START_REF]EPAPS Document No. xxxxxxx for rf amplitudes and phases of optimal control sequences and MATLAB scripts for the optimal control design of these[END_REF] A. Coherence-order and spin-state-selective coherence transfer

For a two-spin, J-coupled spin systems in the context of liquid-state NMR, the internal Hamiltonian may in the high-field approximation be cast as

H 0 = πJ2I z S z , (15)
where I z or S z represent z-components of the spin operators for the I and S spins in the present example coupled through a scalar coupling of size J = 140 Hz. To exemplify transfer between non-Hermitian operators, we assume the initial state to be represented by +1-quantum coherence on the S spin (i.e., ρ 0 = S + ) and to further demonstrate spin-state-selective transfer, [START_REF] Nielsen | [END_REF][55][56][57] we assume that the destination operator is -1-quantum coherence on the I spin with the S spin being in the α-state corresponding to only one of the lines in the J-coupled doublet excite (i.e., C=I -S α ).

In this case of transfer between non-Hermitian operators, we use the target function in Eq.( 6) modified to ensure our target being positive semi-definite. This leads to the modified objective φ and the operator Θ(C):

φ = φ 2 + Tr(U (T )U † (T )), (16)
Θ(C) = U † (T ) + ρ 0 U † (T )C † Tr U † (T )ρ † 0 U (T )C + C † U † ρ 0 Tr U (T )ρ 0 U † (T )C † , (17)
where we, relative to the expression in Eq. ( 11), assumed the scaling factor to be κ ≡ 1 for the sake of simplicity.

Our control fields are represented by the amplitudes (angular frequencies) ω k (t) of xand yphase rf irradiation on the spins I (operators I x (k=1) and I y (k=2)) and S (operators S x (k = 3) and S y (k = 4)). According to unitary bounds on spin dynamics, [58,59] the maximum achievable transfer efficiency in this case is 1. For the optimizations we set the excitation period to T = 7.14 ms (corresponding to 1/J) and the number of pulses to N = 200.

Addressing this specific optimization problem, Figure 1 compares pulse sequences (i.e., I and S spin rf field strengths) obtained on basis of a random initial pulse sequence using the gradient-based optimal control algorithm GRAPE, [19] the monotonic Krotov-based optimal control algorithm of Maximov et al., [38] and the latter combined with smoothing as described in this paper. It is quite evident that the pulse sequences developed using the two former methods display quite significant oscillations, for which the appearance depends very much on the specific optimization. In the present case it looks like the GRAPE algorithm produces a more smooth, albeit still oscillating a lot in the "waves", pulse sequence than the Krotov-based algorithm. Slight variation in parameters, such as the number of steps in the waves, may radically change this pictures and in a later example we will see the opposite picture.

The same variability will be seen upon inspection of the many optimal control NMR pulse sequences presented in literature so far. What is clear, however, is that the Krotov-based algorithm (as selected in this paper, but it could just as well be GRAPE) combined with frequency-truncation smoothing leads to much smoother sequences which may easier be analysed analytically and which put less demands on the spectrometer hardware upon practical implementation. Apart from this the sequences display only relatively modest variations with respect to rf power consumption (root-mean-square (RMS) average rf powers (I and S spin values separated by /) of 141/71 Hz for GRAPE, 92/91 Hz for Krotov, and 12/102 Hz for smoothing Krotov) and are essentially identical with respect to coherence transfer efficiency (99% of the theoretical maximum for all methods).

While rigid statements on the optimization speed and convergence require a much more thorough analysis (to be presented elsewhere), examples like the one in Fig. 1 provide the following crude estimate. Smooth Krotov is computation-speed wise quite similar to GRAPE, while the original Krotov approach is somewhat faster (in the order of a factor 3-5, although this number highly depends on the specific optimization). Out of 1000 optimizations based on random initial pulse sequences only 80-90 % of the GRAPE and Krotov optimizations led to sequences with more than 90% of the nominal transfer efficiency, while all sequences in this specific case passed this limit for the smooth Krotov approach. The great benefit of smooth Krotov is that it provides smooth pulse sequences, and the benefit of both Krotov variants is that they enable optimization with much more coarse time dicretization (longer and fewer pulses) than the GRAPE algorithm due to its fundamentally different optimization strategy.

Figure 2 gives snapshots of pulse sequences throughout optimization using the smoothing Krotov-based optimal control algorithm starting out from a random pulse sequence (with maximum amplitude of 100 Hz). The snapshots illustrate gradual adoption of the optimal sequence to a smooth appearance. It is evident that already after 5 iterations, the control fields on the I spin almost vanishes, leaving the major action to the S spin control fields for the remaining optimization into the final pulse sequence. The final pulse sequence have quite low rf-power consumption, due to the reducing effect of the running cost (λ 1 = 10 -4 s 2 rad -2 , λ 2 = λ 1 for both the I and S spins), and the rf fields vary smoothly as an effect of the smoothing algorithm. The progress of the cost function and its constituents (the overall functional (cost), the transfer efficiency, and the running cost (penalty)) throughout the optimization is illustrated in Fig. 3 which also provide a numerical demonstration of monotonic convergence. It is evident that the functional converges to a value which, when considering the subtractive term from the running cost, is equal to or close to the theoretical limit after a number of iterations where the efficiency (final cost) and penalty (running cost)

have displayed some exchanging oscillations before converge to the optimal values.

B. Optimal control for satellite and central transition excitation in 23 Na MRI

Optimal control mediated experiment design is by no means restricted to spin-1/2 nuclei or systems of these.

Many challenging optimization examples may be found for NMR spectroscopy or MRI in concern of quadrupolar nuclei, as recently demonstrated by optimal control pulse sequences for improved multiple-quantum magic-anglespinning NMR [60] and for selective excitation of the central transition of 23 Na (spin I = 3/2) in presence of residual quadrupole couplings for magnetic resonance imaging purposes. [START_REF] Lee | [END_REF][35][36] The latter application may be very important for 23 Na MRI of, e.g., cartilage where the sodium concentration may be considered a reporter for disorders and degradation. [START_REF] Lee | [END_REF]61] In such applications, the ability to separate 23 Na ions with large and very small (vanishing) residual quadrupolar couplings is regarded important as they represent different populations of relevant ions.

In this section, we demonstrate the use of the smoothing Krotov-based optimal control algorithm to design pulse sequences which selective excites the central transition (-1 2 , 1 2 ) or the satellite (-3 2 ,-1 2 or 1 2 , 3 2 ) transitions of 23 Na ions characterized by a residual quadrupole coupling frequency of ω Q /2π = 60 Hz. In this case the size of the quadrupole coupling and the rf irradiation fields are comparable (often referred to as the intermediate regime) implying that experiment design by standard analytical means is not straightforward.

The optimization involves the secular, high-field-approximated first-order Hamiltonian

H = ω Q 2 (3I 2 z -I(I + 1)) + ω 1 (t)I x + ω 2 (t)I y , (18)
where ω Q is quadrupole coupling frequency while ω 1 (t) and ω 2 (t) are control rf fields corresponding to the spin operators I x and I y , respectively (all frequencies in angular units). The nature of the optimization will be a standard state-to-state transfer with the final cost expressed by Eq. ( 5) with the destination operator C representing x-phase coherence on either the central transition or the satellite transitions. The initial state corresponds to the thermal equilibrium ρ 0 = I z .

Figure 4 illustrates the performance of optimal control pulse sequences designed to enhance the central transition (in this case leading to efficient suppression of the satellite transitions although not requested in the cost function used for the optimization) or excite the satellite transitions while suppressing the central transition (also here the suppression was not specified in the cost function) relative to a hard (infinitely strong) 90 o pulse exciting both types of transitions. The simulated spectra clearly reveal that (i) the central-transition selective excitation optimal control sequence offers a sensitivity enhancement, relative to standard single-pulse excitation, by the maximal factor of 1.5 as discussed previously by Jerschow and coworkers, [32] while ensuring reasonable suppression of the satellite transitions and (ii) the pulse sequence designed to suppress the central transition lead to satellites with unaltered intensity while the central transition is almost removed.

The optimal control pulse sequences providing selective excitation of the central transition and the satellite transitions are shown in Figs. 5A and5B, respectively. The sequences have a overall length of 2.25/f Q (with the peak splitting f Q related to the quadrupole coupling as

f Q = 3ω Q /2π
) and accommodates 200 pulses of equal length. The central transition excitation pulse sequence is characterized by a RMS average rf power of 45 Hz and excites the central transition with an efficiency of 99% and the satellites have been suppressed to an efficiency of 1.4% (percentages relative to the theoretical maximum). The satellite transition sequence is characterized by an RMS rf power of 54

Hz, with the satellites excited with an efficiency of 99% and the central transition suppressed 72%.

It is clear from both sequences that the optimal pulse sequence is found in the regime ω rf ∼ ω Q for which experiment design by standard analytical means is exceedingly difficult. It is also evident that the smoothing algorithm used during optimization leads to smooth sequences offering easier implementation on conventional MRI instrumentation, than the corresponding sequences we obtained using previous optimal control formulations. The latter aspect is demonstrated in Fig. 6A by central transition-selective excitation sequences (corresponding to Fig. 5A) obtained using GRAPE or in Fig. 6B using our previous Krotov-based optimal control software. The latter pulse sequences involved 500 pulses for GRAPE and 200 pulses for the Krotov-based approach to facilitate comparison with the pulse sequences earlier proposed by Jerschow and coworkers [START_REF] Lee | [END_REF] for GRAPE. In the present implementation, the GRAPE-derived sequence displays much wilder oscillations than the Krotov-based sequence.

C. Broadband optimization

In practical applications of optimal control theory for experiment design, it is often desirable to optimize the pulse sequences to be robust toward variation in one or more spin system parameters, introducing the concepts of broadband or bandselective excitation as described previously in the context of GRAPE [24] and methods.

Such requests may readily be incorporated into our optimal control algorithm through a modified cost function

J(ω) = κTr(U (T )U † (T )) + i Tr(C † U i (T )ρ 0 U † i (T )) -λ k T 0 ω 2 k (t)dt, (19)
where each propagator U i (T ) corresponds to a specific condition, such as different chemical shifts, couplings, or orientations of the sample relative to the magnetic field (in solid-state NMR), where U is a common propagator of the system. The additional targets in the functional Eq. ( 19) corresponding to the broadband extension demands a modification of the condition of a monotonic convergence and as concequence changing the form of Eq.( 12). This leads to a modification of the target function f j (ω ′ j ) in Eq.( 12) where the part involving propagators and Lagrange matrices should be replaced by a summation

• • • conditions i A i U ′ i,j-1 B i,j-1 A † i . . . (20)
with the propagator U i and adjoint matrix B i computed for each individual condition when the pulse sequence ω k is the same for all propagators.

To demonstrate the application of broadband optimization, we repeat our optimization of 23 Na MRI central transition enhancement experiments for a range of quadrupole coupling around ω Q /2π = 60 Hz with a deviation of ±20

Hz. This leads to a pulse sequence with a central transition excitation profile as function of the quadrupole coupling frequency as illustrated in Fig. 7, revealing very good excitation in proximity of ω Q /2π = 60 Hz as requested in the optimization. We note that it is obviously possible to specify specific regions of excitation and no excitations by entering additional conditions in the target function and algorithm.

VI. CONCLUSION

In conclusion, we have presented a smoothing monotonically convergent optimal control algorithm for efficient design of pulse sequences for magnetic resonance applications. The proposed frequency-truncation algorithm has been incorporated into a Krotov-based optimal control procedure and demonstrated for a few of NMR examples where it leads to much smoother optimal pulse sequences than obtained using previous methods. We note that previous methods may eventually lead to smooth sequence by themselves, in particular for simple optimizations initiate the optimization. However, in the most general case optimal control typically leads to pulse sequences with significant oscillations in phase and amplitude. In such cases our new algorithm offers a remedy to incorporate smoothness in the optimization. The smooth sequences improves the possibilities for obtaining theoretical insight for the optimal sequences and facilitates implementation on typical spectrometer hardware. We anticipate that the methods will find widespread applications for design of experiments within liquid-and solid-state NMR spectroscopy, MRI, and dynamic nuclear polarization.

VII. 4) and ( 16); for transparency, we have subtracted the correction terms in Eq. ( 16) which otherwise would add a constant contribution to the functional in all points), the green the efficiency φ 2 , and blue the penalty cost (see text). All ordinates have been scaled by a factor T r{CC † }.

3 . 4 .

 34 Set ǫ = +∞, ℓ = 0. While ǫ > T ol do a. Do forward propagation and search new rf field ω ℓ+1 according to the procedure in Sec. III b. Apply the frequency truncation sub-algorithm with regularization to find α ℓ that preserves the monotonicity, i.e.: J((1

Fig. 1 .

 1 Fig.1. (Color online) Pulse sequences for spin-state-and coherence-order-selective S + → I -S α coherence transfer in a J-coupled (J=140 Hz) heteronuclear two-spin system designed using optimal control based on the GRAPE algorithm (first row), the original monotonic Krotov-based algorithm (second row), and the latter combined with frequency-truncation smoothing (third row). The red and green lines correspond to xand y-phase rf control fields for the spins I (left column) and S (right column), respectively.

Fig. 2 .

 2 Fig. 2. (Color online) Representative pulse sequences obtained throughout the optimization in Fig. 1, illustrating the gradual adoption of the pulse sequence to provide optimum S + → I -S α coherence transfer with modest rf-power consumption and smooth rf variation. The red and green lines correspond to xand y-phase control fields for spin I while blue and pink lines represent xand y-phases of the S-spin control fields.

Fig. 3 .

 3 Fig. 3. (Color online)The dependence of the functional J(ω), the efficiency φ 2 , and the running cost (penalty) throughout the iteration steps of a smoothing monotonic Krotov-based optimal control design of the pulse sequence for S + → I -S α coherence transfer, shown and analysed in Figs.1 and 2. The red curve represents the functional J(ω) (defined by Eqs. (4) and (16); for transparency, we have subtracted the correction terms in Eq. (16) which

Fig. 4 .

 4 Fig. 4. (Color online) Simulated 23 Na NMR spectra of sodium with a residual quadrupole coupling frequency of ω Q /2π = 60 Hz excited using a standard 90 o hard pulse (red line), the optimal control pulse sequence in Fig. 5A designed to excite exclusively the central transition (green line), and the pulse sequence in Fig. 5B designed to excite the satellite transitions only (blue line).

Fig. 5 .

 5 Fig. 5. (Color online) Optimal control pulse sequences for 23 Na MRI, which for the case of a residual quadrupole coupling of 60 Hz offer selective and enhanced excitation of the central transition (A) or excitation of the satellite

Fig. 6 .

 6 Fig. 6. (Color online) Optimal control pulse sequence for excitation of the central peak of the quadrupole spectrum of 23 Na (A) (parameters as described in relation to Fig. 4) obtained using the GRAPE algorithm and (B) the original Krotov based algorithm. The red and green curves represent xand y-phase control fields. The length of pulse sequences in both cases is 12.5ms.

Fig. 7 .

 7 Fig. 7. (Color online) (a) Excitation profile (normalized) for the central transition as function of the quadrupole coupling frequency for a pulse sequence optimized using our smooth variant of the Krotov algorithm and specifying "broadband" excitation for a range of quadrupole coupling frequencies in proximity of 60 Hz. (b) The length of the pulse sequence is 12.5 ms, the RMS average rf power is 170 Hz, and the average transfer efficiency in the range of quadrupolar couplings between 40 and 80 Hz is 1.89 (corresponding to 95 % of the theoretical maximum).

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ACKNOWLEDGMENTS

This work was supported by the EU 6th Framework Program in terms of the BIODNP research infrastructure project. Support from the Danish National Research Foundation, the Danish Centre for Scientific Computing, and the Kavli Institute for Theoretical Physics, University of California, Santa Barbara (NCN) is acknowledged. GT and JS were partially supported by the "Agence Nationale de la Recherche" (ANR), Projet Blanc C-QUID number BLAN-3-139579. We thank Christoffer Laustsen for discussion on 23 Na MRI.