Bolton
email: philbolton2000@yahoo.co.uk

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Whilst reporting to SHOT has historically been voluntary, a number of governing bodies and laboratory accreditation organisations have made reporting compulsory.

Reporting of serious adverse events and reactions related to blood transfusion is now a legal requirement, following implementation of the Blood Safety and Quality Regulations in 2005 (1) for which the Medicines and Healthcare products Regulatory Agency is the competent authority. This means that SHOT has near universal up take by all UK blood transfusion laboratories. SHOT produces an annual report where a break down of the previous year's incidents are presented and discussed. New and continuing trends are identified and recommendations on how to improve practice are then suggested. 4.2% of red cells are transfused to patients less than 18 years old and 1.7% to children less than 1 year old (2). Looking at SHOT data for these groups, there is a disproportionate number of adverse reactions in these groups, which can be estimated at 18:100,000 transfusions for children less than 18 years of age and 37:100,000 in 3) The adult rate is only 13:100,000. So why do children have a higher incidence of adverse reactions during transfusion of blood products with nearly three times the adult rate in infants? To answer this, the figures need to be broken down.

SHOT classifies incidents reported into 12 different categories. (table 1)

Incorrect blood component transfused

This was the most common category in both adults and children, but it was particularly common in children, making up more than 80% of reports (3).

Special requirements not met

Many children have very specific requirements. CMV negative blood is required for all children less than one year of age. Blood must be irradiated to kill any remaining white cells if the recipient is immunocompromised (for example Di George syndrome).

Omission of CMV negative or irradiated blood were the most common mistakes seen in this group. Failure to use irradiated blood in an immunocompromised child can put them at risk of graft verses host disease. This is an extmemly rare, but universally fatal complication of transfusion.

Another example occurred (4) when an exchange transfusion was undertaken in a neonate with blood which was more than 5 days old. This caused in an electrolyte imbalance leading to serious arrhythmia. Methylene blue treated plasma must be used in all children born after 1996 to minimise the theoretical risk of transmission of nvCJD. This problem can be minimised by electronically flagging up all children born after this date to minimise the risk of this occurring.

Administration errors

There are a number of unique problems seen in children with regard to identification.

Neonates may not have been named and young children can not confirm their identity.

Twins have the same date of birth and surname and can often be cared for in the same neonatal unit. SHOT 1999 illustrated this problem where the wrong identical twin neonate was transfused in error (6). There were 2 cases where blood samples were taken simultaneously from a mother and baby and mislabelled. In one case it resulted in a group O baby transfused with group A blood (7) Another problem peculiar to children is the potential to treat members of the same family together. An example of the potential for confusion occurred when 2 brothers who were both to be transfused for thalassaemia major swapped beds for a prank between the checking of the blood and it's administration and received each others blood. (8) Lab errors These remain most common in the neonatal group reflecting the complicated nature of transfusion in this age group. For example incompatible transfusions were reported (4,9) with a number of neonates because of maternal antibodies which had not been detected in the baby's sample. The mother's sample had mistakenly not been taken into account during the crossmatch process.

Inappropriate / Unnecessary transfusion

While blood can be transfused on the strength of a mistaken haemaglobin to an adult or a child there are a number of situations where children are at particular risk of over transfusion. The importance of clear prescription and communication can be illustrated by 2 examples. In the first a prescription for 50ml of blood to be given over 4 hours was misunderstood as 50ml/hour of blood for 4 hours, resulting in morbidity (6). In the second case a 2-year old child was under shared care for treatment of a malignancy.

The central hospital requested that the shared care hospital transfuse 2 paedipacks of red cells without mentioning the number of mls required. Two adult units were transfused resulting in a very high haemoglobin which required venesection. Both of these cases highlight the importance of having a standardised method to prescribe blood which is unlikely to be misunderstood and to always use mls as the volume rather than units.

Handling and Storage Errors

The strict requirements for neonatal transfusion are also seen in this category with half of the errors reported in this category involving neonates. For example the blood for exchange transfusion in a neonate was more than 5 days old at the time it was given. (10)

Acute Transfusion Reactions

This is the most common non-error related category, making up around a quarter of all cases reported. The overall incidents in children is similar to adults. Neonates are rarely seen in this group. There are a number of possible explanations for this.

Neonates are relatively immunosupressed and therefore may not mount such an obvious reaction to blood products. Also neonates requiring blood are commonly unwell and a transfusion reaction may be thought to be due to other factors. Platelets have made up about half of all acute transfusion reactions. This increased in 2008 to 72% of the total (10). Since platelet transfusions make up a very small amount of the total units transfused, it is apparent that acute transfusion reactions are at least 5 -6 times more likely in this group (11) The high incidence of reaction to platelets is not seen in adult practice where red cell transfusion most commonly cause acute reactions. The reactions reported in children also tended to be more severe than in

Autologous transfusion problems

This category relates mostly to intra-operative cell salvage and re-transfusion. Only 2 of the 28 reports in 2008 relate to children (10), but it is mentioned here because of the potential increase in use in paediatric surgery in the future. The 2 children reported were teenagers having orthopaedic surgery. There were technical difficulties with the cell salvage equipment resulting in abandoning of the technique without any harm to either child. Looking at the adult data from this year, the most common report (5 cases) was of hypotension during re-infusion of the blood. Common factors in all of these reports were:- This is the category that most members of the general public (and some clinical staff) worry about, after the problems of hepatitis B & C, HIV and more recently concern of the possibility of nvCJD. The fact is, however this is a rare event, making up 2% of all paediatric reports. In the last 2 years (2007-8) there has been only one report of a bacterial infection which was the result of a platelet transfusion. There were no viral infections reported in the last 4 years in children. The last recorded episode was in 2004 when a 2-year old girl was infected with hepatitis B from a regular platelet donor during the very early acute phase of infection. (SHOT 2004) When considering the possibility of long incubation infections occurring in children, it is important to remember that the transfusion recipients may be living with the consequences for many decades into the future. While nvCJD remains vanishingly rare, with no reports of infection in children, new, unexpected and novel infections may occur in the future, so vigilance is vital.

•

Transfusion Related Acute Lung Injury (TRALI)

This is a rare condition in children with only 2 reports in 2008 (10). Both were teenagers. It may be an important category however in anaesthesia and intensive care medicine, since it is probably under reported. It is hard to diagnose, since an isolated deterioration in lung function can commonly being attributed to other causes in sick and unstable patients. This may also be the reason why there are so few reports of TRALI in neonates, although this may be due to an impaired immune response in this group.

Recommendations

SHOT makes recommendations at the end of every annual report which reflects any changes in practice or trends which are developing. The reports also comment on previous recommendations and discusses how they have progressed. SHOT 2007 reemphasised the importance of Laboratory staff being appropriately trained and aware of the special requirements for children under the age of 16. Also the importance of ward staff who prescribe blood products being suitably trained and experienced to calculate the appropriate volume of blood required for a transfusion in a child. This prescription must always be in mls, never in units.

The British Committee for Safety in Haematology has produced formal guidelines for transfusion of neonates and older children, which are based on the evidence from SHOT (12).

Discussion

The majority of adverse reactions in children were error related. Often the error was from the issuing laboratory, but errors from clinical staff were an important factor in many incidents. Anaesthetists treat a wide range of children requiring transfusion support who are often unstable with a complex clinical picture. In this environment, changes in the clinical condition of a child may be due to many factors. Suspicion that a drop in blood pressure may be the result of an acute transfusion reaction or Neonates are a particular challenge for anaesthetists. Knowledge of their specific transfusion requirements will help to prevent inappropriate and potentially dangerous practice from occurring. This is particularly true when a specific diagnosis or treatment is being managed, such as Di-George syndrome or ECMO. Good communication with the issuing blood bank is vital to ensure that the correct blood is issued in these very complicated cases.

Conclusion

The advances in surgical and medical care for children, such as cardiac surgery and oncology have been paralleled by the advances in transfusion practice, without which the clinical work would have been impossible. Continued reporting of adverse events and reactions to SHOT will contribute to a growing body of evidence on transfusion hazards that can influence future transfusion safety strategies and improve education.

Improved quality of practice is already suggested by the increasing number of reports over the last 10 years while the number of serious events has declined. All of this should result in improved patient outcome. Whilst reporting to SHOT has historically been voluntary, a number of governing bodies and laboratory accreditation organisations have made reporting compulsory.

Reporting of serious adverse events and reactions related to blood transfusion is now a legal requirement, following implementation of the Blood Safety and Quality Regulations in 2005 (1) for which the Medicines and Healthcare products Regulatory Agency is the competent authority. This means that virtually all UK blood transfusion laboratories report to SHOT. SHOT produces an annual report where a break down of the previous year's incidents are presented and discussed. New and continuing trends are identified and recommendations on how to improve practice are then suggested. 4.2% of red cells are transfused to patients less than 18 years old and 1.7% to children less than 1 year old (2). Looking at SHOT data, there is a disproportionate number of adverse reactions in these groups, which can be estimated at 18:100,000 transfusions for children less than 18 years of age and 37:100,000 in those under 1 year (3). The adult rate is only 13:100,000. So why do children have a higher This was the most common category in both adults and children, but it was particularly common in children, making up more than 80% of reports (3).

Special requirements not met

Many children have very specific requirements for blood products. There are a number of reasons for this. In the first year of life, children have an immature immune system and are at particular risk of certain blood born infections. A number of congenital syndromes are associated with the requirement for neonatal surgery. If the diagnosis has not been confirmed, a precautionary policy is required with blood transfusion.

CMV negative blood is required for all children less than one year of age. Blood must be irradiated to kill any remaining white cells if the recipient is immunocompromised (for example Di George syndrome). Omission of CMV negative or irradiated blood were the most common mistakes seen in this group. Failure to use irradiated blood in an immunocompromised child can put them at risk of graft verses host disease. This is an extremely rare, but universally fatal complication of transfusion. Some procedures, which only occur in the pre-natal or neonatal period require very specific blood products, for example intra-uterine transfusion or neonatal exchange transfusion. Failure to adhere to these strict criteria has caused significant electrolyte imbalance in infants leading to serious arrhythmia (4). Children born after 1996 in the UK have in theory avoided exposure to new variant Creutzfeldt-Jakob Disease (nvCJD) in the food chain. In an attempt to avoid the potential for transmission of this infection, methylene blue treated plasma must be used in all children born after this date. Many of these problems can be minimised by using an electronic flagging system based on the patient's date of birth. This recommendation was made in the (5).This group of errors however still accounts for the biggest difference between children and adults.

Administration errors

There are a number of unique problems seen in children with regard to identification.

Neonates may not have been named and young children can not confirm their identity.

Twins have the same date of birth and surname and can often be cared for in the same neonatal unit. SHOT 1999 illustrated this problem where the wrong identical twin neonate was transfused in error (7). There were 2 cases where blood samples were taken simultaneously from a mother and baby and mislabelled. In one case it resulted in a group O baby transfused with group A blood (8). Another problem peculiar to children is the potential to treat members of the same family together. An example of the potential for confusion occurred when 2 brothers who were both to be transfused for thalassaemia major swapped beds for a prank between the checking of the blood and it's administration and received each others blood (9).

Laboratory errors

These remain most common in the neonatal group reflecting the complicated nature of transfusion in this age group. For example incompatible transfusions were reported (4,10) with a number of neonates because of maternal antibodies which had not been detected in the baby's sample. The mother's sample had mistakenly not been taken into account during the crossmatch process. While blood can be transfused on the strength of an inaccurate haemoglobin to an adult or a child, there are a number of situations where children are at particular risk of over transfusion. The importance of clear prescription and communication can be illustrated by 2 examples. In the first case, a prescription for 50ml of blood to be given over 4 hours was misunderstood as 50ml/hour of blood for 4 hours, resulting in morbidity (7). In the second case a 2-year old child was under shared care for treatment of a malignancy. The central hospital requested that the shared care hospital transfuse 2 paedipacks of red cells without mentioning the number of mls required. Two adult units were transfused resulting in a very high haemoglobin, which required venesection. Both of these cases highlight the importance of having a standardised method to prescribe blood which is unlikely to be misunderstood and to always use mls as the volume rather than units.

Inappropriate / Unnecessary Transfusion

Handling and Storage Errors

The strict requirements for neonatal transfusion are also seen in this category with half of the errors reported involving neonates. For example the blood for exchange transfusion in a neonate was more than 5 days old at the time it was given (11).

Acute Transfusion Reactions

This is the most common non-error related category, making up around a quarter of all cases reported. The overall incidents in children is similar to adults. Neonates however are rarely seen in this group. There are a number of possible explanations for this. Neonates are relatively immunosupressed and therefore may not mount such an obvious reaction to blood products. Also neonates requiring blood are commonly unwell and a transfusion reaction may be thought to be due to other factors. (11). Since platelet transfusions make up a very small amount of the total units transfused, it is apparent that acute transfusion reactions are at least 5 -6 times more likely in this group (12). The high incidence of reaction to platelets is not seen in adult practice where red cell transfusion most commonly cause acute reactions. The reactions reported in children also tended to be more severe than in adults with 52% in 2008 being classed as severe allergic / anaphylactic type compared to 20% of adult reactions (11).

Autologous Transfusion Problems

This category relates mostly to intra-operative cell salvage and re-transfusion. Only 2 of the 28 reports in 2008 relate to children (11), but it is mentioned here because of the potential increase in use in paediatric surgery in the future. The 2 children reported were teenagers having orthopaedic surgery. There were technical difficulties with the cell salvage equipment resulting in abandoning of the technique without any harm to either child. Looking at the adult data from this year, the most common report (5 cases) was of hypotension during re-infusion of the blood. Common factors in all of these reports were:-

•

Transfusion Transmitted Infections

This is the category that most members of the general public (and some clinical staff) worry about, after the problems of hepatitis B & C, HIV and more recently concern of the possibility of nvCJD. The fact is, however this is a rare event, making up 2% of all paediatric reports. In the last 2 years (2007-8) there has been only one report of a bacterial infection which was the result of a platelet transfusion. There were no viral infections reported in the last 4 years in children. The last recorded episode was in 2004 when a 2-year old girl was infected with hepatitis B from a regular platelet donor during the very early acute phase of infection (4).

When considering the possibility of long incubation infections occurring in children, it is important to remember that the transfusion recipients may be living with the consequences for many decades into the future. While nvCJD remains vanishingly rare, with no reports of infection in children, new, unexpected and novel infections may occur in the future, so vigilance is vital.

Transfusion Related Acute Lung Injury (TRALI)

This is a rare condition in children with only 2 reports in 2008 (11). Both were teenagers. It may be an important category however in anaesthesia and intensive care medicine, since it is probably under reported. It is hard to diagnose, since an isolated deterioration in lung function can commonly being attributed to other causes in sick and unstable patients. This may also be the reason why there are so few reports of TRALI in neonates, although this may be due to an impaired immune response in this group.

More than 95% of the cases in the last ten years have been caused by plasma or platelets.

Recommendations

SHOT makes recommendations at the end of every annual report which reflects any changes in practice or trends which are developing. The reports also comment on previous recommendations and discuss how they have progressed. SHOT 2007 reemphasised the importance of Laboratory staff being appropriately trained and aware of the special requirements for children under the age of 16. Also the importance of ward staff who prescribe blood products being suitably trained and experienced to calculate the appropriate volume of blood required for a transfusion in a child. This prescription must always be in mls, never in units.

The British Committee for Standards in Haematology has produced formal guidelines for transfusion of neonates and older children, which are based on the evidence from SHOT [START_REF] Boulton | Transfusion Guidelines for Neonates and Older Children[END_REF].

Discussion

The majority of adverse reactions in children were error related. Often the error was from the issuing laboratory, but errors from clinical staff were an important factor in many incidents. There were a number of categories where children were more commonly seen than adults. These include failure to meet special requirements, failure to correctly identify the patient and difficulties with crossmatching. While acute transfusion reactions are equally common in children and adults, platelet reactions are more common in children. Neonates are a particular challenge for anaesthetists. Knowledge of their specific transfusion requirements will help to prevent inappropriate and potentially dangerous practice from occurring. This is particularly true when a specific diagnosis or treatment is being managed, such as Di-George syndrome or ECMO. Good communication with the issuing blood bank is vital to ensure that the correct blood is issued in these very complicated cases.

The advances in surgical and medical care for children, such as cardiac surgery and oncology have been paralleled by the advances in transfusion practice, without which the clinical work would have been impossible. Continued reporting of adverse events and reactions to SHOT will contribute to a growing body of evidence on transfusion hazards that can influence future transfusion safety strategies and improve education.

Improved quality of practice is already suggested by the increasing number of reports over the last 10 years while the number of serious events has declined. All of this should result in improved patient outcome.

  of Transfusion (SHOT) is the United Kingdom's national audit of clinical transfusion practice. It has been collecting data since 1996 and in the first ten years it collected 3239 reports of adverse reactions relating to the transfusion of labile blood components (from a total of 23.4 million). SHOT is funded by the four United Kingdom blood transfusion services and is affiliated to the Royal College of Pathologists. Strategic direction is provided by a steering group which has wide representation from UK Royal Colleges and professional bodies including the Royal College of Anaesthetists and the Intensive Care Society.

  in 2008 being classed as severe allergic / anaphylactic type compared to 20% of adult reactions.(10)

  Use of acid citrate dextrose as anticoagulant• Use of a leucodepletion filter during reinfusion of red cells These patients may have been relatively hypovolaemic and thus more susceptible to vasoactive cytokines present in the blood. All of these episodes were transient, being corrected by stopping the infusion +/-vasopressors. There were no long term sequelae.

9

 9 More than 95% of the cases in the last ten years have been caused by plasma or platelets.

  be the result of a transfusion related acute lung injury is helped when the anaesthetist is aware of what group of children are most at risk and which products most commonly are associated with these conditions.

  Transfusion Practitioner. Royal Hospital for Sick Children, Yorkhill, Dalnair St. Glasgow G3 8SJ. Scotland, UK. Dr Philip Bolton. Consultant Anaesthetist. Royal Hospital for Sick Children, Yorkhill, Dalnair St. Glasgow G3 8SJ. Scotland UK. Correspondence to philbolton2000@yahoo.co.uk of Transfusion (SHOT) is the United Kingdom's national audit of laboratory and clinical transfusion practice. It has been collecting data since 1996 and in the first ten years it collected 3239 reports of adverse reactions relating to the transfusion of labile blood components (from a total of 23.4 million). SHOT is funded by the four United Kingdom blood transfusion services and is affiliated to the Royal College of Pathologists. Strategic direction is provided by a steering group which has wide representation from UK Royal Colleges and professional bodies including the Royal College of Anaesthetists and the Intensive Care Society.

  reactions during transfusion of blood products with nearly three times the adult rate in infants? Also how can awareness of the errors in transfusion of children improve their care during surgery? To answer this, the figures need to be broken down.SHOT classifies incidents reported into 13 different categories. (table1

7

 7 Platelets have made up about half of all acute transfusion reactions. This increased in 2008 to 72% of the total

  Use of acid citrate dextrose as anticoagulant• Use of a leucodepletion filter during reinfusion of red cells These patients may have been relatively hypovolaemic and thus more susceptible to vasoactive cytokines present in the blood. All of these episodes were transient, being the infusion +/-vasopressors. There were no long term sequelae.

11 Anaesthetists

 11 treat a wide range of children requiring transfusion support who are often unstable with a complex clinical picture. In this environment, changes in the clinical condition of a child may be due to many factors. Suspicion that a drop in blood pressure may be the result of an acute transfusion reaction, or worsening ventilation may be the result of a transfusion related acute lung injury is helped when the anaesthetist is aware of what group of children are most at risk and which products most commonly are associated with these conditions.

Pediatric Anesthesia


Kirsten Stiebler

Bettina Gertjerenken

Niklas Teichmann

Christoph Weiss

Spatial two-particle NOON-states in periodically shaken three-well potentials

Keywords: numbers: 05.60.Gg, 03.75.Gg, 03.75.Lm

Few-particle dynamics in a three-well potential are investigated numerically. It is shown that periodically shaking the potential can considerably increase the fidelity of emerging spatial quantum superpositions. Such NOON-states are important for quantum interferometry. If the two particles initially sit in the middle well, the probability to return to this state can distinguish pure quantum dynamics from statistical mixtures. The numeric implementation of decoherence via particle losses shows clear differences from the pure quantum behaviour. A three-well lattice could be an ideal system for experimental realisations.

Introduction

Double-well lattices [1,2] provide an interesting, experimentally accessible tool to investigate quantum effects for atom numbers even down to less than six atoms [3]. Three-well lattices and four-well lattices could be realised either by using subwavelength lattices [4] or by adjusting the technique used in Refs. [3,5] by using more lasers. The number of particles in one of the two wells of a double-well lattice can easily be measured by averaging over all double wells [3]. For a single three-well potential, transport and interaction blockade of cold bosonic atoms have been discussed in Ref. [6]. In a single three-well potential, few particle dynamics could be observed by using the single-site addressability of Refs. [7,8].

One of the fascinating aspects of quantum mechanics are non-classical quantum states like the NOON-states [9]:

|Ψ NOON ≡ 1 √ 2 (|N, 0 + |0, N ) , (1)
where |N -n, n refers, e.g., to N -n particles being on the left and n particles on the right of some barrier but NOON-states can also be investigated in phase-space. These states have also been called "Schrödinger cat"-states, e.g., in Refs. [10,11].

Like the spin-squeezed states of Refs. [12,13], such states are relevant to improve interferometric measurements [14]. Suggestions on how such interesting many-particle quantum superpositions might be obtained can be found, e.g., in Refs. [15,16,17,18,19,20,21,22,23,24,25] and references therein.

The focus of the present paper lies on the numeric creation of spatial two-particle NOON-states in three-well potentials with a focus on experimental signatures to verify that indeed NOON-states rather than statistical mixtures have been created. To achieve the NOON-states via controlled quantum dynamics, we suggest to periodically shake the three-well potential. This approach to enhance tunnelling in a controlled way is based on photon-assisted tunnelling [26,27,28,29,[START_REF] Esmann | This unpublished manuscript was made available to the referees of J. Phys. B[END_REF] which was realised experimentally for Bose-Einstein condensates in optical lattices [31] (for the interpretation of the experimental data see also [32]). Recent research in periodically driven systems include nonlinear Landau-Zener processes [33] and transport of bound pairs [34,35].

The paper is organised as follows: in Sec. 2 we introduce the Hamiltonian used to model the periodically shaken three-well potential as well as our main aim, the two-particle NOON-states. Section 3 shows that for some parameters, the quantum dynamics of the periodically shaken system can be understood within a simplified model originally developed for double wells [START_REF] Esmann | This unpublished manuscript was made available to the referees of J. Phys. B[END_REF] which we adopt for the present situation. In Sec. 4 we introduce the two-particle NOON-states and explain how they can be distinguished from statistical mixtures. Section 5 shows that periodic shaking can considerably increase the fidelity, i.e., the quality of the quantum superpositions. In order to show that the quantum superpositions can indeed be distinguished from statistical mixtures, last but not least decoherence effects are included. This situation is an experimentally realistic initial condition [41]. Experimental investigations would, e.g., be possible in three-well lattices [4] (cf. the double-well lattice potential with less than six atoms per lattice site of Ref. [3]). Periodic shaking is currently established as an experimental tool to control the quantum dynamics [31,32,42,43].

Model system

For the description of ultracold atoms in a three-well potential, the three-site version of a many-particle Hamiltonian [36] originally developed in nuclear physics [37] is used:

Ĥ = -J ĉ † 1 ĉ2 + ĉ † 2 ĉ1 + ĉ † 3 ĉ2 + ĉ † 2 ĉ3 + U 2 3 i=1 ĉ † i ĉ † i ĉi ĉi + 2 [hµ 0 + hµ 1 sin(ωt)] ĉ † 1 ĉ1 -ĉ † 3 ĉ3 (2)
The operator ĉ( †) j annihilates (creates) a boson in well j; J is the hopping matrix element, 2hµ 0 is the tilt between well 1 and well 3 and 2hµ 1 is the driving amplitude. The interaction between a pair of particles in the same well is denoted by U . Quantum dynamics beyond such models has been investigated, e.g., by Refs. [6,38,39,40] ‡. A systematic sketch of the three-well potential is shown in Fig. 1.

A Fock-basis is useful to describe N atoms in the three-well potential:

|ψ N -n-m,n,m = |N -n -m, n, m ≡ |n, m (3)
with m particles in the right well, n particles in the middle well and N -m -n particles in the left well.

The aim of this paper is not to produce a single-particle quantum superposition but a two-particle NOON-state:

|Ψ δ = 1 √ 2 |2, 0, 0 + e iδ |0, 0, 2 . (4)
As will be shown in this paper, periodic shaking helps on the way to achieve this aim. In order to characterise the quality with which this aim is achieved for a given normalised wave-function |ψ(t) ( ψ(t)|ψ(t) = 1), the fidelity,

F δ = | ψ(t)|Ψ δ | 2 , (5)
or more precise the fidelity maximised over all possible angles δ is used:

fidelity = max (F δ , 0 ≤ δ < 2π) . (6)
In order to be a useful flag to indicate a NOON-state, the fidelity has to be larger than 50%; below this value, the state might not be a superposition at all as, e.g., the state |2, 0, 0 would also result in a fidelity of 0.5. Figure 2 displays an example for which the fidelity is well above 50%. The data is plotted as a function of the dimensionless time,

τ ≡ Jt h . (7)
Figure 3 shows that for the timescales of Fig. 2 and a large parameter range, even without shaking, the fidelity can reach values well above 50% (Fig. 3) but it stays again below 90% (Fig. 3). Thus, so far there does not seem to be an advantage of using periodic shaking. However, before showing that periodic shaking can lead to fidelities well above 95%, we show that it is possible to quantitatively understand the behaviour displayed in Fig. 2.

Simplified model

In order to understand the behaviour under periodic shaking, a simplified version of the Hamiltonian (2) can be used. This approach introduced below will be useful for large shaking frequencies and not too long timescales [START_REF] Esmann | This unpublished manuscript was made available to the referees of J. Phys. B[END_REF].

Extending the simplified model of Ref. [START_REF] Esmann | This unpublished manuscript was made available to the referees of J. Phys. B[END_REF] to a three-well potential starts with writing the Schrödinger equation corresponding to the Hamiltonian (2) in the Fock basis (3). Using the short notation of Eq. ( 3), we then introduce the ansatz motivated by the interaction picture

ν, µ|ψ(t) = a ν,µ (t) exp - i h ν, µ|H 0 (t)|ν, µ dt , (8)
with H 0 ≡ lim J→0 H. The tunnelling between Fock states is then included in a system of first-order one-dimensional differential equations for the a ν,µ (cf. Refs. [28,[START_REF] Esmann | This unpublished manuscript was made available to the referees of J. Phys. B[END_REF]).

The time-dependence enters entirely via the exponents introduced in the ansatz (8); it includes terms linear in time (either proportional to the interaction or to the tilt) as well as a term proportional to the driving amplitude for which the time-dependence 6) as a function of dimensionless time τ [Eq. ( 7)] for the experimentally realistic initial condition [41] of two particles sitting in the middle well of a three-well potential. The parameters were chosen such that shaking frequency, interaction and tilt are equal: hω = U = hµ 0 = 6J which corresponds to integer photon-assisted tunnelling [27,29]. The shaking amplitude was chosen to be hµ 1 = 25.2J. It has been optimised with the simplified model (17). Solid grey line: the exact numerical solution of the time-dependent model Hamiltonian (2). Dashed black line: the simplified model which can be solved via exact diagonalisation. While the fidelity is well above 50%, it still remains below 90% (the exact numerics reaches the maximum 0.8852 at τ = 4.787).

is a cos(ωt). This last term can be transferred into a sum of exponentials with linear time-dependence via [START_REF] Abramowitz | Pocketbook of Mathematical Functions[END_REF]

e iz cos(ωt) = ∞ k=-∞ J k (z)i k e ikωt , (9)
where the J k (z) are Bessel functions. This leads to a series of relevant frequencies which for two particles read:

σ k ≡ U h + 2µ 0 -kω (10)
σ k ≡ - U h + 2µ 0 -kω (11)
σ k ≡ 2µ 0 -kω . ( 12
)
The model is still equivalent to the full Schrödinger equation corresponding to Eq. ( 2); for each set of parameters, the maximum is plotted. Similar to the periodically shaken case displayed in Fig. 2, the fidelity stays below 90% (0.892 at U = 1.3J, hµ 0 = -0.5J and τ 15). Higher fidelities will be achieved via periodic shaking (cf. Fig. 4) it now reads:

ih∂ t             a 0,0 a 0,2 a 2,0 a 1,0 a 1,1 a 0,1             =             0 0 0 A 1 (t) 0 0 0 0 0 0 A * 2 (t) 0 0 0 0 A * 2 (t) A 1 (t) 0 A * 1 (t) 0 A 2 (t) 0 0 A 3 (t) 0 A 2 (t) A * 1 (t) 0 0 A * 3 (t) 0 0 0 A * 3 (t) A 3 (t) 0                         a 0,0 a 0,2 a 2,0 a 1,0 a 1,1 a 0,1             (13)
where

A 1 (t) ≡ -J √ 2 k J k 2µ 1 ω i k exp(iσ k t) , (14)
A 2 (t) ≡ -J √ 2 k J k 2µ 1 ω i k exp(i σ k t) , (15)
A 3 (t) ≡ -J k J k 2µ 1 ω i k exp(i σ k t) . (16)
In general, for given parameters of tilt, interaction and shaking frequencies, there will be only one term with slow time-dependence §. Following the reasoning of the § Cases for which the above sums involve two frequencies with opposite sign but similar magnitude are rotating wave approximation [START_REF] Haroche | Exploring the Quantum -Atoms, Cavities and Photons[END_REF], this term should dominate the dynamics for not too large timescales. Defining

σ k ≡ U h + 2µ 0 -k ω ( 17
)
where k is the integer for which | U h + 2µ 0 -kω| reaches its minimum and repeating the same reasoning for σ k and σ k leads to a much simpler version of the matrix in Eq. ( 13) which surprisingly can even be solved via exact diagonalisation (see Ref. [START_REF] Esmann | This unpublished manuscript was made available to the referees of J. Phys. B[END_REF] where even some analytical results could be derived). The derivation of our simplified model explains [cf. Eq. ( 17)] why the two curves in Fig. 2 agree well except for the deviation for larger times.

How to distinguish two-particle NOON-states from statistical mixtures

The main idea is to be able to distinguish quantum superpositions from statistical mixtures via the behaviour concerning the return to the initial state. Why a tilted three-well potential is useful for considerations like that can already be motivated on the single-particle level: On the one hand, despite the tilt the tunnelling processes

|0, 1, 0 → 1 √ 2 (|1, 0, 0 ± |0, 0, 1 ) → |0, 1, 0 (18)
do not involve any change of the potential energy. On the other hand, the individual tunnelling processes

|0, 1, 0 → |1, 0, 0 |0, 0, 1 → |0, 1, 0 . (19)
are related to a change in energy (because of the tilt) and thus both the first and the second tunnelling process of Eq. ( 19) would thus be suppressed. This offers a starting point to distinguish the time-evolution of quantum superpositions from statistical mixtures: On the one hand, tunnelling of the quantum superposition

1 √
2 (|1, 0, 0 ± |0, 0, 1 ) back to the initial state |0, 1, 0 is allowed energetically. On the other hand, for a statistical mixture like = 1 2 (|1, 0, 0 1, 0, 0| + |0, 0, 1 0, 0, 1|) ( 2 0 ) the tunnelling to the state |0, 1, 0 is suppressed for energetic reasons. There is, however, an even stronger reason why the return to the initial state can verify interesting quantum superpositions:

While experimentally measuring the fidelity (6) would either be impossible or at least be a considerable challenge for future experiments, measuring the number of atoms not likely to be covered by this approximation. Furthermore, for the slowly varying term to be really the dominant term, it has to be assumed that its prefactor is not much smaller than that of the other terms in the sum.

The time-dependent model (2) can, in general, only be solved by numerically integrating the timedependent Schrödinger equation. Within this paper, we do this by using the Shampine-Gordon routine [START_REF] Shampine | Computer Solution of Ordinary Differential Equations[END_REF]; exact diagonalisation is a much simpler approach.

in one of the wells is comparatively straight-forward both in an optical superlattice [3] and in a single three-well potential [7,8]. Thus, for the two-particle NOON-state 1 √ 2 (|2, 0, 0 + exp(iδ)|0, 0, 2 ) one could experimentally verify, that with nearly 50% probability both particles are in the left well and with 50% probability both particles are in the right well. In order to distinguish this situation from the statistical mixture, = 1 2 (|2, 0, 0 2, 0, 0| + |0, 0, 2 0, 0, 2|) ,

we will be searching for parameters such that the probability to return to the initial state is high. As the initial state is the Fock state with two particles in the middle well, this is an unambiguous experimentally verifiable signature. It remains to be shown that the return to the initial state at time t 2 -in combination with the 50/50 probability to find both particles either in the left or in the right well at time t 1 < t 2 -really is unambiguous. Using the time-evolution operator Û , one has:

|ψ(t 2 ) = Û (t 2 , t 1 )|ψ(t 1 ) . ( 22
)
Let us consider one particular ideal set of parameters for which the wave-function ψ 1 (t) reaches the perfect cat-state (4) at t = t 1 followed by a probability to return to the initial state at t = t 2 of 100%:

| 0, 2, 0|ψ 1 (t 2 ) | 2 = 1 . (23)
Within this ideal set of parameters, an inaccurate preparation of the wave-function could lead to a return-probability of just x:

| ψ 2 (t 2 )|ψ 1 (t 2 ) | 2 = x , 0 ≤ x ≤ 1 . ( 24
)
The fact that Û † Û = 1 implies:

| ψ 2 (t 1 )|ψ 1 (t 1 ) | 2 = x . (25)
Thus, for our example of a perfect return to the initial state, a fidelity of x at t = t 1 leads to a probability of x to return to the initial state at t = t 2 . In particular, if a NOON-state has a 100% return probability to the initial state at a given time, the corresponding statistical mixture (21) will only manage a return to the initial state with 50% probability at the same point of time ¶. While we will investigate one particular source of decoherence -particle losses -in the next section, other sources of decoherence would also lead to statistical mixtures and thus to a different dynamics. Given the level at which today's experiments with ultra-cold atoms are performed, we can assume that the particles do mostly behave as predicted by the Schrödinger equation without decoherence.

If the return to the initial state is less than the expected 100% (but still well above 50%), this does not necessarily imply that the NOON-state was partially destroyed via decoherence: experimental uncertainties which would average over different solutions of the Schrödinger equation could produce a similar effect. Nevertheless, such a high ¶ To see this, one can use the Schrödinger equation seperately for the time evolution of |2, 0, 0 and |0, 0, 2 . 6) as a function of time for two particles initially sitting in the middle well and hω = 6J, U = 9.1J in an untilted three-well potential (µ 0 = 0) with a shaking amplitude of hµ 1 = 18.3J. Black solid line: numeric solution of the model (2); the fidelity reaches values of 99.67%, well above 90%. Gray solid line: the simplified model (17) does not even predict the correct frequency of the oscillations; taking 15 terms in Eqs. ( 14)-( 16), again leads to a better agreement (dashed black line).

return probability can still be a signature of the NOON-state: It has to be checked if the variations of the parameters still lead to high fidelities at t = t 1 . In an experiment this would lead to probabilities of the order of 50/50 to find the two particles either in the left or in the right well.

NOON-states with fidelities above 90%

To obtain better results than displayed in Figs. 2 and3, three requirements were used for the numeric optimisation of the parameters:

• a fidelity well above 90%.

• a high probability to return to the initial state.

• not too large timescales.

There are parameter-sets with and without tilt that fulfil these conditions. One example is shown in Figs. 4 and5. The approach via the simplified model (17) only works if more than one frequency is included in Eqs. ( 14)-( 16) (see Fig. 4). At first glance, the standard deviation displayed in Fig. 5 might render it difficult to distinguish the behaviour including decoherence from the ideal quantum case. However, for the accuracy with which the average can be measured, the standard error is relevant which is smaller by a factor of 1/ √ M for M measurements. Taking M 100 this would thus reduce the error by a factor of 10. Our theory so far ignores decoherence. For the present system, atom losses via scattering with back-ground atoms are an important decoherence mechanism (cf. Refs. [22,[START_REF] Kohler | [END_REF]). To include this into our model, we use a method called piecewise deterministic processes, PDP in Ref. [START_REF] Breuer | The Theory of Open Quantum Systems[END_REF], which is based on Refs [START_REF] Dalibard | [END_REF]50]. The main idea is to use quantum dynamics for the time-evolution in combination with atom losses for which the points of time of the atom losses are chosen via random numbers [START_REF] Breuer | The Theory of Open Quantum Systems[END_REF]51]. The results displayed in the middle panel of Fig. 5 are averaged over many sets of random numbers. Atoms are lost with a loss-rate α (cf. Appendix A),

d dt N (t) = -α N (t) , (26)
corresponding to an exponential decay of the average number of particles, which for the two particles reads:

N (t) = 2 exp(-αt) . (27)
The middle panel of Fig. 5 uses α = α 1 with α 1 = J/(30h); for the timescales of the double-well lattice experiment [3], the fidelity reaches its maximum at 9.7 . . . 16.1 ms and thus α ≈ 1/36 . . . 1/22 kHz. In order to be able to observe the return to the initial state with 1% accuracy, we require the average number of particles ( 27) to be at least 99% of the initial value up to a timescale for which pure quantum evolution predicts a nearly perfect return to the initial state (32 ms). This results in a lifetime of the particles of 1/α 2 ≈ 3 s. This realistic lifetime of a single particle in the lattice lies two orders of magnitude below the lifetime required to produce NOON-states of, say, 100 particles according to the scheme of Ref. [22]. The lower panel of Fig. 5 shows what would happen if the quantum superposition was turned into a statistical mixture at the time for which the fidelity reaches its maximum: although now there still are two particles involved in the simulation, the average occupancy of the middle well would remain a factor of two below the values reached for pure quantum evolution.

In order to reproduce the generation of NOON-states shown in Figs. 4 and5, an experiment would require a control of both interaction (U/J) and driving amplitude hµ 1 of the order of several per cent, the tilt 2hµ 0 should be zero to about 0.04J/h in order to reach fidelities above 0.95 + . In an experiment, there might be problems in realising a periodic shaking which is exactly zero for times t < 0 and perfectly sinusoidal for times t > 0. To show that this is not necessarily a problem, Fig. 6 shows what happens if the shaking amplitude is switched on and off with T settl as the characteristic timescale:

B(t) = 2hµ 1 sin(ωt + φ) 1 -exp -2πt T settl 1 -exp -2π(T -t) T settl , t ≤ 0 ≤ T ; ( 2 8 )
+ The requirements on the accuracy of the parameters were obtained by calculating both the fidelity and the return to the initial states at the time defined by the ideal case (rather than using the maximum fidelity/maximum return for each parameter set). the shaking now includes a phase φ in sin(ωt + φ). Plotted is the mean deviation from the case without any increase of the amplitude:

mean | n 2 -n 2 id | 2 ≡ 1 T T 0 | n 2 -n 2 id | 2 dt ( 29
)
where n 2 id is the model (2).

Conclusion

To conclude, we have shown via numerical investigations that high-fidelity-two-particle NOON-states can emerge in three-well potentials. There is, in particular, an easily observable difference between statistical mixtures and perfect NOON-states: only the latter would return to the initial states. For realistic conditions, decoherence would be present in such experiments, which would destroy the quantum superpositions. Here, the proposed scheme might offer the possibility to measure the timescales of decoherence. Three-well lattices could be used to realise such experiments -produced either via a superposition of lasers similar to the double-well lattices of Refs. [3,5] or by using subwavelength lattices [4]. The number of atoms per three-well lattice can be controlled precisely via the Mott-insulator used to load the lattice [3]. Such a setup would allow to measure the average number of particles of one of the three lattice sites in a single experiment by averaging over all three-well potentials. Furthermore, by focusing on one single three-well potential within the three-well lattice via techniques similar to Refs. [7,8] and repeating the experiments many times, even more precise measurements could be possible.

Figure 1 .

 1 Figure1. The three-well potential with initially two atoms in the middle well. This situation is an experimentally realistic initial condition[41]. Experimental investigations would, e.g., be possible in three-well lattices[4] (cf. the double-well lattice potential with less than six atoms per lattice site of Ref.[3]). Periodic shaking is currently established as an experimental tool to control the quantum dynamics[31,32,42,43].

Figure 2 .

 2 Figure 2. Fidelity (6) as a function of dimensionless time τ [Eq.(7)] for the experimentally realistic initial condition[41] of two particles sitting in the middle well of a three-well potential. The parameters were chosen such that shaking frequency, interaction and tilt are equal: hω = U = hµ 0 = 6J which corresponds to integer photon-assisted tunnelling[27,29]. The shaking amplitude was chosen to be hµ 1 = 25.2J. It has been optimised with the simplified model(17). Solid grey line: the exact numerical solution of the time-dependent model Hamiltonian (2). Dashed black line: the simplified model which can be solved via exact diagonalisation. While the fidelity is well above 50%, it still remains below 90% (the exact numerics reaches the maximum 0.8852 at τ = 4.787).

Figure 3 .

 3 Figure 3. Fidelity in a two-dimensional projection as a function of both interaction U and tilt hµ 0 without any shaking. To obtain the plotted data, the model (2) was solved numerically for two particles initially in the middle well and times τ with 0 ≤ τ ≤ 50.for each set of parameters, the maximum is plotted. Similar to the periodically shaken case displayed in Fig.2, the fidelity stays below 90% (0.892 at U = 1.3J, hµ 0 = -0.5J and τ 15). Higher fidelities will be achieved via periodic shaking (cf. Fig.4)

Figure 4 .

 4 Figure 4. Fidelity (6) as a function of time for two particles initially sitting in the middle well and hω = 6J, U = 9.1J in an untilted three-well potential (µ 0 = 0) with a shaking amplitude of hµ 1 = 18.3J. Black solid line: numeric solution of the model (2); the fidelity reaches values of 99.67%, well above 90%. Gray solid line: the simplified model(17) does not even predict the correct frequency of the oscillations; taking 15 terms in Eqs. (14)-(16), again leads to a better agreement (dashed black line).

Figure 5 .

 5 Figure 5. Average number of particles in the second well as a function of time for the same parameters as in Fig. 4. Upper panel: thick solid line: n 2 as predicted by quantum dynamics obtained by numerically solving the model (2); at τ 13.36 there are hardly any particles in the middle well ( n 2 = 0.026) and at τ 26.69 the particles have returned to the middle well ( n 2 = 1.9919). Dashed lines: n 2 ± ∆n 2where ∆n 2 is the standard deviation. Middle panel: thick solid line: n 2 as predicted by a combination of quantum dynamics and particle losses(26) for α = J/(30h). Black dashed lines: n 2 ± ∆n 2 where ∆n 2 is the standard deviation. Lower panel: n 2 as predicted by quantum dynamics obtained by numerically solving the model (2) if the quantum superposition turns into a statistical mixture at the point of highest fidelity (τ max = 13.356). The required accuracy of the parameters is discussed above Eq.(28).

Figure 6 .

 6 Figure 6. Mean square deviation (29) for T = 100h/J in two-dimensional projection as a function of both settling time T settl [see Eq. (28)] and initial phase. All other parameters as in Fig 4.T drive ≡ 2π/ω is the oscillation period. Contrary to experiments like[52], in our case the initial phase does not play any role.

‡ Reference[3] demonstrates that the Hamiltonian (2) can indeed be valid to describe the dynamics of few particles in few well systems.

Acknowledgements

We thank M. Holthaus for his support and H.-P. Breuer, M. Esmann and F. Schreck for discussions. BG acknowledges funding by the Studienstiftung des deutschen Volkes. NT acknowledges funding by the University of Oldenburg. CW acknowledges support by C. & C. Hettich in initiating this project.

Appendix A. Lindblad equation to model particle losses

A derivation of the Lindblad Master equation can be found, e.g., in Ref. [START_REF] Breuer | The Theory of Open Quantum Systems[END_REF]. Often, a ( †) is the annihilation (creation) operator of a photon in mode ω but atom losses can be described by the same approach:

where ˆ is the density matrix. We assume that particles only are lost, thus κ + = 0 and α ≡ κ -and thus derive a rate equation [START_REF] Haroche | Exploring the Quantum -Atoms, Cavities and Photons[END_REF] by using the fact that the photon number distribution p n (t) is related to the density matrix via

The rate-equation thus reads:

Using this rate-equation, it is straight-forward [START_REF] Haroche | Exploring the Quantum -Atoms, Cavities and Photons[END_REF] to derive the exponential decay of the numbers of particles given in Eq. (26). As explained at the beginning of Sec. 5, we use piecewise deterministic processes [START_REF] Breuer | The Theory of Open Quantum Systems[END_REF] (cf. [START_REF] Dalibard | [END_REF]50]) to model the atom losses; in between loss events, the Hamiltonian dynamics of Eq. ( 2) is used. Further theoretical investigations of atom losses can be found, e.g., in Ref. [51].


K Khanbekyan
email: karen.khanbekyan@gmail.com

G Bevilaqua

A Khanbekyan

E Mariotti

A Papoyan

L Moi

A Phenomenological Model For Collisional Coherence Transfer In Optically Pumped Atomic System

Keywords: numbers: 32.70.-n, 32.70.Jz, 33.40.+f, 33.57.+c

We consider a dual Λ-system under double laser excitation to investigate the possibility of indirect coherence transfer between atomic ground states through an excited state. The atomic system is excited by a frequency modulated pump laser and probed by a low-power cw laser. All the decoherence mechanisms are discussed and taken into account. Adjustment of parameters of the two radiations aimed at maximization of coherence transfer is addressed. The study can help to understand phenomena as collisional transfer of coherence and can find application in the experimental realization of atomic sensors.

Introduction

Optical pumping technique serves as a powerful tool for spin polarizing an atomic sample since early fifties [Kastler(1950)], see also the review [Happer(1972)]. Modulating a laser beame is a well known issue since the pioneristic work of [START_REF] Series | Theory of the modulation of light in optical pumping experiments[END_REF]] and can be profitable used to observe resonant effects. These experimental methods recently attracted renewed attention due to its various applications in many different fields and its potential importance in the development of atomic probes of highest sensitivity.

Optical atomic magnetometers constitute a significant example. In fact sensitivities as high as 22 fT/

√

Hz have been obtained [Shah(2010)], that, although are not comparable with those of SQUID sensors (≈ 10 -1 fT/ √ Hz), allow for the possibility of miniaturizing the sensor volume [Schwindt(2004)] [ Ledbetter(2008)] or of getting rid of cryogenics as they work at room temperature or higher. Recent experiments demonstrated the possibility of low field NMR [Belfi(2009)]. In last paper a laser beam is frequency modulated to produce a pulsed excitation of atoms. This creates an atomic polarization that, due to an external magnetic field, precesses at Larmor frequency. The aim of that work was to build a new self oscillating optical magnetometer based on non linear magneto optical rotation and to open a way to new applications and ideas. For example, shining another transition of the atomic sample by using a cw probe laser we expect to find a transferred modulated behaviour on the probe. In this way the possibility of coherent transfer could be demonstrated.

The goal of this paper is to study this opportunity. In fact, we have developed a model representing the atomic sample as a simplified Λ-type level scheme consisting of an excited level and two pairs of ground state levels. Atoms are excited by a pulsed pump and by a cw probe laser: when the frequency of pumping laser is modulated and synchronized with the Larmor precession, the optical pumping efficiency resonantly enhances [Acosta(2006)] [ Dehmelt(1957)] [ Bell(1957)]. Because the two optical transitions share the same excited state, coherence transfer between the ground state levels is allowed. On the other hand coherence and population transfer is also possible by means of relaxation effects and it is interesting to investigate the importance of the latter on the predicted signal.

The paper is organized as follows. In Section 2 the model is widely discussed. In Section 3 we present and discuss the numerical results related to the optimization of the lasers parameters. Finally Section 4 contains the conclusions.

Model

Let us consider the simplified atomic model sketched in Fig. 1.

= |F = 3, M F = 2 , |3 = |F = 3, M F = 3 . Similarly |4 = |F = 4, M F = -4 , |5 = |F = 4, M F = -3 .
The pump beam, of frequency ω 1 , is tuned to the transition |2 → |1 with variable detuning δ 1 (t); the probe beam, of frequency ω 2 , is tuned to the transition |5 → |1 with detuning δ 2 . The Rabi frequencies are respectively Ω P r and Ω P u . Ω L i are the Larmor frequencies of the |2 → |3 (i = 1) and |4 → |5 (i = 2) transitions. We assume that the spectral widths of the lasers are narrower than the atomic spectral linewidth. The pump laser frequency is modulated with a rectangular function signal whose period T can be adjusted with respect to the Larmor one

T = κ 2π Ω L 1 . (1)
The time dependence of pump detuning is shown in Fig. 2. The frequency modulation of the pump laser produces synchronous Zeeman optical pumping and a macroscopic magnetization build-up. If the modulation frequency is not synchronized with the Larmor frequency, the magnetizations associated with subsequent light pulses are randomly oriented and average to zero in few pulses (as shown in Fig. 8). On the other hand the same behavior is observed when the duty cycle of the modulating square wave is large, enforcing the idea that the pumping should not significantly disturb the A Phenomenological Model For Collisional Coherence Transfer In Optically Pumped Atomic System4 .

Figure 2: Typical rectangular signal function used to modulate the pump laser frequency.

In the calculations a duty-cycle of ∼ 2 % is assumed.

spins during the Larmor precession.

Hamiltonian

The total Hamiltonian is equal to the sum of the unperturbed term H 0 and the atomfield interaction addendum H 1 (t),

H = H 0 + H 1 (t) ( 2 )
In the basis of the |j (j = 1, . . . , 5) states, the Hamiltonian H 0 reads (see Fig. 1)

H 0 =        E e 0 0 0 0 0 -E g 0 0 0 0 0 -E g 0 0 0 0 0 E g 0 0 0 0 0 E g        . (3)
The atom-field interaction Hamiltonian H 1 (t) consists of a sum of the probe laser-atom interaction term H P r (t), the pump laser-atom interaction term H P u (t), and the magnetic field-atom interaction term H B (t):

H 1 (t) = H P r (t) + H P u (t) + H B , (4)
where

H B = h Ω L 1 2 |2 3| + h Ω L 2 2 |4 5| + H.c., (5)
A Phenomenological Model For Collisional Coherence Transfer In Optically Pumped Atomic System5

and in the rotating wave approximation the coupling to the optical fields reads

H P u (t) = hΩ P u e -i(ω 1 t+Φ(t)) |1 2| + H.c., (6)
H P r (t) = hΩ P r e -iω 2 t |1 5| + H.c., (7)
Here ω 1 + Φ(t) is the instantaneous frequency of the pump laser (see Fig. 2).

Transforming to the rotating frame specified by the unitary operator

U (t) = |1 1| + (|2 2| + |3 3|) e (iω 1 t+Φ(t)) + (|4 4| + |5 5|) e iω 2 t ,
we get the following expression for the transformed Hamiltonian H = U † HU + iU † U :

H = h         0 Ω P u 0 0 Ω P r Ω P u δ 1 (t) Ω L 1 2 0 0 0 Ω L 1 2 δ 1 (t) 0 0 0 0 0 δ 2 Ω L 2 2 Ω P r 0 0 Ω L 2 2 δ 2         , (8)
where (assuming h ≡ 1 from now on)

δ 1 (t) = (E e + E g ) -(ω 1 + Φ(t)) (9) δ 2 = (E e -E g ) -ω 2 (10)

Relaxation

Proper description of relaxation processes in atomic systems requires the density matrix formalism, whose evolution is governed by optical Bloch equations:

∂ρ(t) ∂t = -i[ H, ρ(t)] + L D ρ(t) = L H ρ(t) + L D ρ(t), (11)
where L D represents the relaxation processes, which are taken into account in the following standard form:

(L D ) ij,km = -δ ik δ jm Γ ij (1 -δ ij ) + δ ij δ km (Γ k→i - n Γ i→n δ ki ). ( 12
)
Here Γ i→j is population transition rate from level i to level j while Γ ij = Γ ji are the coherence damping parameters. The rates Γ i→j describes spontaneous decay from the excited state to the ground states, population transfer in the ground states both between hyperfine levels with rates γ L 24 , γ L 25 , γ L 34 , γ L 34 and between the Zeeman sublevels with rates γ L 23 γ L 45 . So we model the population transfer as follow

Γ i→j =        0 Γ/4 Γ/4 Γ/4 Γ/4 0 0 γ L 23 γ L 24 γ L 25 0 γ L 23 0 γ L 34 γ L 35 0 γ L 24 γ L 34 0 γ L 45 0 γ L 25 γ L 35 γ L 45 0        , ( 13
)
where we assumed equal branching ratio for the spontaneous emission from the excited state, which has the natural linewidth equal to Γ. The Γ ij matrix contains the A Phenomenological Model For Collisional Coherence Transfer In Optically Pumped Atomic System6 damping due to the population transfer via the rates G i = j Γ i→j , as well as a "phenomenological" part Γ phen ij which describes dephasing processes unable to change the populations

Γ phen ij =        0 0 0 0 0 0 0 γ C 23 γ C 24 γ C 25 0 γ C 23 0 γ C 34 γ C 35 0 γ C 24 γ C 34 0 γ C 45 0 γ C 25 γ C 35 γ C 45 0        . ( 14
)
The total matrix is

Γ ij = G i + G j 2 + Γ phen ij . ( 15
)
Here γ C 24 , γ C 25 , γ C 34 , γ C 34 and γ C 23 , γ C 45 are the coherence relaxation rates between hyperfine and Zeeman levels respectively.

Various physical mechanisms affect the hyperfine and Zeeman relaxation rates. The most important contributions are related to collisions with the cell walls, with the buffer gas atoms and with atoms of the same kind giving rise to spin-exchange. A wide and detailed account can be found in [Vanier(1989)]. Here we are interested in those rates as phenomenological parameters only and we do not pursue further any microscopic identification. Moreover we do not take into account the Doppler broadening because the presence of buffer gas changes the atomic motion from balistic to diffusive (this is the main reason of using the buffer gas).

The density matrix time evolution was performed numerically using standard tools to solve initial value problems. We assumed that at t = 0 the atoms start to interact with the external fields. After a transient which is related to the initial state ρ(0), the system settles in the time-dependent stationary state. We have carefully checked that the stationary state is unrelated to ρ(0) (as expected on general ground); additionally we found that the final state is reached faster starting the simulation with all the atoms in the excited level.

Results

We decided to focus our attention on the imaginary part of ρ 51 , information which is directly comparable with the absorption of the probe beam. A typical time evolution of [ρ 51 (t)] is shown in Fig. 3.

At early time intervals [ρ 51 (t)] shows a transient time dependence, then after many cycles of pump laser but very rapidly on the laboratory time scale, it reaches the stationary regime, which is of interest for our study. Vertical lines in Fig. 3 correspond to beginning of time intervals where the pump laser frequency fits the resonance (see Fig. 2). The absorption then oscillates at approximately the Rabi frequency. When the pump laser goes out of resonance the absorption get driven by the "free evolution" governed by the magnetic field.

A Phenomenological Model For Collisional Coherence Transfer In Optically Pumped Atomic System7 In other words one can think that, when the pump laser is out of resonance, the probe laser moves atoms from level |5 to level |1 so that the levels |2 , |3 are populated by spontaneous emission from level |1 while level |5 (and accordingly |4 ) are depopulated. The process is inverted turning on the pump laser thus repopulating levels |4 and |5 via spontaneous decay. However a proper quantitative analysis is more complicated than this simple picture because of the incoherent population transfer between the ground states. We are interested in studying the behaviour of the Larmor's oscillation amplitude A (see inset in Fig. 3) as a function of the various model parameters. Our numerical findings are reported in the following subsections.

Time evolution of ρ 51 (t) for different Larmor frequencies

Time dependence of [ρ 51 (t)] for different values of Larmor frequency (i.e. for the different values of external magnetic field) is shown in Fig. 4. Here assuming that Ω L 1 = ±Ω L 2 we specifically think that the ground levels refer to the ground hyperfine states of alkali atoms.

The [ρ 51 (t)] amplitude A takes a higher value for the case of Ω L = 0.025Γ as compared with Ω L = 0.01Γ.

Dependence of [ρ 51 (t)] amplitude on Rabi frequency of pump laser

As shown in Fig. 5, for small Ω P u the amplitude of [ρ 51 (t)] approaches zero, then increases and eventually saturates. In fact the more the probe laser is resonant the better the process is enhanced. Secondly at higher values of Ω P u , the atoms are "repumped" in |4 and |5 efficiently from |2 and |3 .

In both circumstance the amplitude of absorption coefficient grows up, arriving at its saturation value.

A Phenomenological Model For Collisional Coherence Transfer In Optically Pumped Atomic System8 It is possible to have an roughly estimate of the saturation intensity of the pump laser using the textbook [Foot(2002)] formula

I sat = π 3 hc λ 3 τ (16)
which is exactly valid for a two level system. In our case for the resonance transition of Cesium (λ = 852 nm, lifetime τ = Γ -1 = 30 ns) the saturation intensity is I sat = 1.3 mW corresponding to Ω P u = 0.1Γ, which is in good agreement with the behaviour shown in Fig. 5 3.3. Dependence of ρ 51 (t) amplitude on Rabi frequency of probe laser

Dependence of [ρ 51 (t)] amplitude on Rabi frequency of probe laser for different Rabi frequencies of pump laser (Ω P u = 0.02Γ, . . . , 0.08Γ) is shown in Fig. 6. The plot corresponds to off-resonance (δ 2 = 1Γ) frequency of the probe laser (Fig. 2).

A When the Rabi frequency of probe laser is zero, the amplitude of [ρ 51 (t)] is zero too, and all atoms are moved to |4 and |5 levels by the pump laser action. With increasing of Ω P r , the amplitude A increases too, reaching a maximum and then starts to decrease. The results can be explained thinking that the probe feels the precessing magnetization M 2 related to the |4 and |5 levels, (namely M 2x ∝ ρ 45 + ρ 54 , M 2y ∝ i(ρ 45 -ρ 54 ), M 2z ∝ ρ 44 -ρ 55 ). Thus for small Ω P r the signal increases as expected reaching a maximum. However when Ω P r increases, the probe is more and more effective in shifting the atoms to levels |2 and |3 , thus reducing the amplitude of M 2 and eventually destroying it.

In Fig. 7 the behaviour of A with respect to κ is shown. The value κ = 1 is missing from the picture because it was impossible to locate in a precise way the maximum and minimum positions of [ρ 51 (t)]. In fact in such a case the Larmor oscillation starts together with the Rabi oscillation (see Fig. 3) leading to unreliable results for A. For this reason we started the picture from κ = 2 value.

At κ = 2 i.e., when the pump laser is modulated with a frequency equal to twice Larmor frequency, A takes the maximum value. In this case the frequency of pump laser is tuned to resonance exactly at the moment when the atomic spin completes one period of damped precession around the external magnetic field. When the frequency of pump laser is switched on to |2 → |1 with κ not close at integer values, the [ρ 51 (t)] amplitude decreases to zero, since the coherence in the ground levels gets destroyed and the asynchronous pumping is not able to restore the magnetization. The amplitude increases again when κ approaches integers restoring the picture of synchronous pumping.

The above line of reasoning is confirmed comparing the time dependence of [ρ 51 (t)] for κ = 1.6 and κ = 2 as shown in Fig. 8.

Dependence of [ρ 51 (t)] on relaxation rates

In this section we will investigate the dependence of [ρ 51 (t)] amplitude on the longitudinal and coherence relaxation rates of the hyperfine and the Zeeman levels. This study is done at the optimum conditions for obtaining the maximum value of absorption amplitude, namely, Ω P u = 0.08Γ, Ω P r = 0.1Γ, δ 2 = 1Γ.

In Fig. 9 we present the dependence of A on population relaxation rates, for the case of off-resonance probe laser detuning (δ 2 = 1Γ) and the following choice of the relaxation parameters

γ L 24 = 2 γ L 25 = 0.1 γ L 23 , γ L 34 = 4/3 γ L 35 = 0.3 γ L 23 , (17)
γ L 45 = γ L 23 ,
which "advantages" the state |4 with respect to the state |5 . However we have found that different, but comparable, choices of these parameters give qualitatively comparable results.

As is seen from the upper part of Fig. 9, A, for small values of Ω P r , does not vanish when the relaxation is zero, since in this case the atoms are moved by pump laser from level |2 to level |1 followed by spontaneous decay to |5 and |4 . As a result, one can see the almost undisturbed Larmor oscillation of ρ 51 (t) on |5 → |1 transition. When the longitudinal relaxation rate increases, the amplitude rapidly decreases indicating that the relaxation processes are very effective in destroying the atomic magnetization.

A Phenomenological Model For Collisional Coherence Transfer In Optically Pumped Atomic System11 On the other hand, for higher values of Ω P r and zero relaxation, the atoms are continuously "pumped off" thus depleting the levels |4 and |5 . Accordingly the Larmor oscillations vanish. Increasing the relaxation the amplitude A increases reaching a maximum and then decreases. It seems that the parameter γ L 23 drives a twofold dynamics. For small values the relaxation contrasts the pumping effect of Ω P r , in fact the higher the value of γ L 23 the more the atoms that incoherently "arrive" in the ground states |4 and |5 , ready to be pumped out by the probe laser. This regime keeps on until γ L 23 reaches a value where the dephasing effects become overwhelming and every magnetization is gradually destroyed, giving a vanishing amplitude A. In Fig. 10 is shown the behavior of A when the Zeeman population relaxation is turned off.

From the picture inspection it seems that incoherent population exchange between the hyperfine multiplets is needed to observe appreciable signals. In fact when γ L is zero no Larmor oscillations are predicted. The explanation is as before: the probe A Phenomenological Model For Collisional Coherence Transfer In Optically Pumped Atomic System12 2,3 with the choice of the other γ outlined in 17 and for different values of Ω P r . The cases with (Ω P r = 0.005Γ, 0.01Γ, 0.03Γ, 0.05Γ, 0.075Γ, 0.1Γ) are magnified 10 times for clarity. In the upper part is shown the initial region, while in the lower part the full picture is displayed.

P u = 0.08 Γ, δ 2 = 1Γ, γ L 24 = 0.2 γ L , γ L 25 = 0.1 γ L , γ L 34 = 0.4 γ L , γ L 35 = 0.1 γ L , while γ L 23 = γ L 45 = 0.
As before, different but comparable choices of these parameters give the same qualitative results. The Larmor frequencies are Ω L 1 = Ω L 2 = 10 -2 Γ. laser, in the stationary state, depletes in a very efficient way the states |4 and |5 . On the contrary with relaxation, some population is supplied to the |4 and |5 states and then probed by the laser. Interestingly enough, the calculated amplitude shows a saturation effect with increasing relaxation, demonstrating that the Zeeman relaxation is responsible of the signal destruction (see Fig. 9). This conclusion is confirmed also by Fig. 11 where the only non-zero parameters are γ L 2,3 = γ L 4,5 .

A Phenomenological Model For Collisional Coherence Transfer In Optically Pumped Atomic System13

Conclusions

We have investigated the interaction of a Λ-type five-level atomic system (atomic vapor in a buffered cell) with two, pump and probe, laser beams, in the presence of an external, time independent magnetic field. The pump laser frequency is modulated in time by a rectangular function synchronized with the Larmor precession. The probe laser absorption shows the Larmor oscillation too and we characterized the amplitude of these oscillations with respect to the model parameters.

We found that the system behaves as expected with regard to the lasers parameters, but an interesting effect is put in evidence when one considers the influence of the relaxation. In fact we have shown that an amount of relaxation, in some circumstances, it is needed in order to keep the signal at good level. Concerning the application of this system as a magnetometer, the sensitivity limit as in other atomic magnetometers is set by the contrast of the observed spectral features, being the signal proportional to the narrow peak amplitude and the noise (assuming a shot noise limited condition) proportional to the background, this value would depend on the specific values of the relaxation rates. As an example, the contrast can be visualized in the inset of Fig 3, as the ratio between A and the average steady state value of [ρ 51 (t)]. Summarizing we have shown that an incoherent population transfer in atomic levels plays an important role.

Figure 1 :

 1 Figure1: Schematic diagram of a five-level Λ-type atomic system, which consists of excited level |1 and two ground state levels with magnetic sublevels |2 , |3 and |4 , |5 . The atomic system is driven by two laser fields, with δ 1 being the pump laser detuning and δ 2 the probe laser detuning. Γ i (i = 2, . . . , 5) is the spectral width of |1 → |i transition.

Figure 3 :

 3 Figure 3: Time dependence of [ρ 51 (t)]. Inset shows the steady state dependence, where A is the oscillation amplitude.

Figure 4 :Figure 5 :

 45 Figure 4: Time dependence of [ρ 51 (t)] for Ω P u = 0.08Γ, Ω P r = 0.3Γ, Ω L = 0.01Γ (dashed line), Ω L = 0.025Γ (solid line).

Figure 7 :

 7 Figure 7: [ρ 51 (t)] amplitude versus the modulation period of pump laser.

Figure 8 :

 8 Figure8: ρ 51 (t) dependence on time when Ω P u = 0.08Γ, Ω P r = 0.3Γ.

Figure 9 :

 9 Figure 9: Amplitude of absorption coefficient on |5 → |1 transition versus γ L2,3 with the choice of the other γ outlined in 17 and for different values of Ω P r . The cases with (Ω P r = 0.005Γ, 0.01Γ, 0.03Γ, 0.05Γ, 0.075Γ, 0.1Γ) are magnified 10 times for clarity. In the upper part is shown the initial region, while in the lower part the full picture is displayed.

Figure 10 :

 10 Figure 10: Dependence of A from "hyperfine" relaxation rate for different values of the probe laser Rabi frequency as shown in the inset. The other relevant parameters are ΩP u = 0.08 Γ, δ 2 = 1Γ, γ L 24 = 0.2 γ L , γ L 25 = 0.1 γ L , γ L 34 = 0.4 γ L , γ L 35 = 0.1 γ L , while γ L 23 = γ L 45 = 0.As before, different but comparable choices of these parameters give the same qualitative results. The Larmor frequencies are Ω L 1 = Ω L 2 = 10 -2 Γ.

Figure 11 :

 11 Figure 11: Behavior of A with respect to the Zeeman population relaxation. The lasers parameters and Larmor frequencies are as before. The only non-zero relaxation parameters are γ L 2,3 = γ L 4,5 .

  Phenomenological Model For Collisional Coherence Transfer In Optically Pumped Atomic System9

					Pu 0.02
		4			Pu 0.04
	A arb. units	1 2 3			Pu 0.08 Pu 0.06
		0
		0	0.05	0.1	0.15
				Pr

Figure 6: Dependence of [ρ 51 (t)] amplitude A on probe laser intensity for different values of pump laser Rabi frequency.

Acknowledgments

The authors are grateful to Y.Dancheva and V.Biancalana from the Magnetometer group of Department of Physics of Siena University for numerous discussions and suggestions and a critical reading of the manuscript. The authors would like to thank D.Sarkisyan, A Phenomenological Model For Collisional Coherence Transfer In Optically Pumped Atomic System14

Yu.Malakyan and A.Kirakosyan for valuable discussions.