Vicens Marti

Joan Garcia-Picart

Jordi Balcells

John Wiley

Sons

CORONARY STENTING AFTER FAILURE OF CONSERVATIVE TREATMENT FOR SPONTANEOUS CORONARY DISSECTION. USEFULNESS OF THE INTRAVASCULAR ULTRASOUND

Keywords: intravascular ultrasound, coronary stenting, coronary dissection

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Spontaneous coronary artery dissection is a rare cause of acute coronary syndrome and sudden death. Prevalence is highest in the peripartum or postpartum period in young women with no cardiovascular risk. It has occasionally been related to intensive physical exercise, cocaine abuse, collagen vascular diseases and antiphospholipidic syndrome. Spontaneous coronary artery dissection most commonly affects the left main coronary artery, the left anterior descending coronary artery (LDA), or both.

Prognosis is related to early diagnosis 1,2 . We describe a woman who had spontaneous coronary artery dissection of the LAD artery in whom angina recurred after initial conservative treatment. She was successfully treated with stent implantation. We emphasise the usefulness of the intravascular ultrasound study (IVUS) before stent implantation.

CASE REPORT

A 52-year-old woman was referred to our coronary care unit with intensive chest pain during moderate effort, associated with autonomic symptoms of 20 minutes' duration, and an ECG showing ST segment elevation of 2 mm in v 2 to v 4 precordial leads. She had a history of smoking, mild obesity, and arterial hypertension treated with bisoprolol. She had recently lost 5 kg in about two weeks as the result of a rigorous hyperproteic diet. When the patient arrived at our center she was asymptomatic and the electrocardiogram was normal. The coronary angiogram showed a diffuse luminal narrowing of the LAD artery from below the origin of the first diagonal to the apical segment (Figure 1,A). Diffuse coronary dissection was diagnosed in the LAD artery.

The right coronary and circumflex arteries were normal. Creatine phosphokinase was 142 U/L and troponin was T 0.118 µg/L. Conservative treatment (atenolol, nitroglycerine, clopidogrel, acetyl-salicylic acid, atorvastatin and intermediate doses of enoxaparin) was adopted because the patient was stable. Three days later, she presented a recurrent episode of angina with ST elevation of 2 mm, extending to leads v 5 -v 6. A second coronary angiogram showed persistence of the diffuse narrowing of the distal LAD artery and occlusion at the apical segment (Figure 2). Before performing an IVUS, we administered bivalirudin at doses of 0.75 mg/kg/weight followed by a perfusion of 1.5 mg/kg during the procedure. After placing an angioplasty catheter and passing a 0.014" guide-wire to the LAD artery, an IVUS using an Atlantis SR Pro 40 MHz coronary imaging catheter (Boston Scientific, Freemont, Ca) was performed to establish correct positioning of the guide-wire in the true lumen and to assess the extension of dissection. No hematoma was seen in the left main or in the proximal or medium segments of the LAD artery (Figure 2, slice 1). A large hypodense image observed about 3 mm below the origin of the diagonal artery was suggestive of an intramural hematoma likely due to recent bleeding (Figure 2, slice 2). A dense extraluminal image suggested an organized hematoma extending through the distal segment with internal hypodense areas (Figure 2, slice 3-4). Since the patient had angina and ST elevation we decided to stent the LAD artery. Stenting was started distally, progressing upwards. Six overlapping conventional stents were implanted: Vision 2,25x18 mm (Abbot Vascular, Santa Clara, Ca); Vision 2,75x12 mm; Driver 2,75x18 mm (Medtronic Inc, Minneapolis, MN), Driver 3.0x24 mm, Driver 3x18 mm, Driver 3.25x30 mm. Balloons were inflated to 15-20 atm, including the overlapping stent segments and flow was reestablished (Figure 1,B). The chest pain disappeared and the ECG normalized. Creatine phosphokinase was 807 U/L and T troponin was 1.8 µg/L. Echography showed mild hypokinesis with a global left ventricular ejection fraction of 60%. Laboratory tests were performed to rule out associated connective tissue disorders and vasculitis. No etiologic cause of the dissection was found. At discharge, in addition to oral nitroglycerin and atenolol, we prescribed clopidogrel 75 mg/day for one year and acetyl-salicylic acid 100 mg/day indefinitely. At 6-month follow-up the myocardial perfusion study using a photon emission computed tomography (SPECT) showed a focal apical necrosis without signs of ischemia in the remaining segments. At 18-month follow up the patient was free of ischemic symptoms and a conventional stress test was negative. Due to the rarity of this event, there is no consensus and optimal treatment remains a challenge. Treatment should be individualised, taking into account the clinical situation, dissection extension, degree of luminal compression, myocardium at risk, multivessel or left main artery involvement and underlying atherosclerotic disease. Conservative treatment with beta-blockers, antiplatelet aggregation and anticoagulation may resolve the hematoma and prevent thrombosis 1 . This approach would be indicated in stable patients.

Optional therapies include coronary artery bypass and stenting. Patients with underlying atherosclerotic disease, multiple vessel dissection or involvement of the left main artery could benefit from an aortocoronary bypass [3][4][5] . Coronary stenting is an increasingly used approach as it rapidly provides lumen permeability 4,5 . It is a valid option in unstable patients with ST elevation with angina and in stable patients with a focal dissection. Stenting has also been successful in patients with left main artery involvement 6 and multi-vessel disease 7 . The inconvenience of stenting is that hematoma may spread, increasing the need for further stenting 8 . To avoid this complication in the present patient we implanted the stents in an upward direction, from the distal to the proximal segment as far as the healthy arterial segment. Stenting may also cause occlusion of the lateral branch due to the adventitial distribution of the hematoma in the arterial wall. Late complications of stenting are restenosis and lack of stent apposition once the hematoma has been reabsorbed, especially when small diameter stents have been implanted. Use of bare-metal and drug-eluting stents has been reported 5 , 9 but the restenosis rate is unknown. Since our patient was asymptomatic and spontaneous reabsorption of the hematoma was likely, conservative However, three days later the ST segment again became elevated and stent implantation was required. We decided to implant bare-metal stents instead of drugeluting stents in view of the positive results reported in the literature and because of the risk of later thrombosis with drug-eluting stents 5,9,10 . IVUS was useful to plan the therapeutic strategy. It differentiated luminal narrowing due to intramural hematoma from a spasm or a diffuse atherosclerotic lesion. It allowed us to assess the extension of the dissection and helped to clarify whether the left main and the proximal LDA artery were also involved, thereby limiting stent implantation. Since coronary angiography provides only a luminal image, stent size calculation is difficult, especially in long dissections. IVUS in this setting allows accurate calculation of vessel size. In coronary dissection there is a risk of inadvertently wiring the false lumen. IVUS assures that the guide-wire is placed in the true lumen, yielding safe stent implantation. In the present case, IVUS helped us to localize the entrance towards the false lumen and to understand the mechanism of recurrence of angina. Distal LAD artery occlusion was seen at the apical level in the second coronary angiogram. IVUS suggested that the entrance was a few millimeters beyond the origin of the first diagonal artery and that occlusion of the distal LAD may have been responsible for the sudden re-bleeding into the false lumen on the third day, causing downwards expansion of the hematoma. This possibility indicates that close surveillance is needed in patients receiving conservative treatment.

In summary, the present case describes the failure of conservative therapy in a woman with spontaneous coronary dissection, highlighting the relevance of IVUS in diagnosis, and illustrating the good results with stent implantation. Prognosis is related to early diagnosis 1,2 . We describe a woman who had spontaneous coronary artery dissection of the LAD artery in whom angina recurred after initial conservative treatment. She was successfully treated with stent implantation. We emphasise the usefulness of the intravascular ultrasound study (IVUS) before stent implantation.

CASE REPORT

A 52-year-old female was referred to our coronary care unit with intensive chest pain during moderate effort, associated with autonomic symptoms vegetatism of 20 minutes' duration, and an ECG showing ST segment elevation of 2 mm in v 2 to v 4 precordial leads. She had a history of smoking, mild obesity, and arterial hypertension treated with bisoprolol. She had recently lost 5 kg in about two weeks as the result of a rigorous hyperproteic diet. When the patient arrived at our center she was asymptomatic and the electrocardiogram was normal. The coronary angiogram showed a diffuse luminal narrowing of the LAD artery from below the origin of the first diagonal to the apical segment (Figure 1

DISCUSSION

The pathogenesis of spontaneous coronary artery dissection is related to a coronary artery disruption that causes a false lumen with an intramural hemorrhage. It progressively compresses the true lumen, leading to myocardial ischemia. Etiology is uncertain, but mechanisms such as disruption of the vasa vasorum, eosinophilic periadventitial inflammation, focal vasculitis, cystic medial necrosis and plaque rupture disruption in patient with atherosclerosis disease have been proposed. Diagnosis is usually made by coronary angiography in the setting of an acute coronary syndrome.

Due to the rarity of this event, there is no consensus and optimal treatment remains a challenge. Treatment should be individualised, taking into account the clinical situation, (e.g. persistent angina, ST elevation), angiographic dissection extension, degree of luminal compression caused by the hematoma, myocardium at risk, multivessel or left main artery involvement and underlying atherosclerotic disease. Successful C Conservative treatment with beta-blockers, antiplatelet aggregation and anticoagulation may promote resolution resolve the hematoma and prevent thrombosis 1 . This approach would be indicated in stable patients.

Coronary artery bypass or stenting are two optional therapies. Optional therapies include coronary artery bypass and stenting [3][4][5] . Patients with underlying atherosclerotic disease, multiple vessel dissection or involvement of the left main artery could benefit from an aortocoronary bypass. However, anastomosis of the graft on a dissected coronary artery is sometimes technically difficult 3 . Coronary stenting is an increasingly used approach as it rapidly provides lumen permeability 4,5 . It is a valid option as a primary election treatment in unstable patients with ST elevation with angina and in stable patients with a focal dissection. Stenting has also been successful used successfully in patients with left main artery involvement 6 and multi-vessel disease 7 .

The inconvenience of stenting is that hematoma may spread, increasing the need for further stenting [START_REF] Chue | Spontaneous coronary artery dissection and the role for percutaneous coronary intervention: to treat or not to treat?[END_REF] . To avoid this complication in the present patient we implanted the stents in an upward direction, from the distal segment to the proximal segment as far as the healthy arterial segment. Stenting may also cause occlusion of the lateral branch due to the adventitial distribution of the hematoma in the arterial wall. Late complications of stenting are restenosis and lack of stent apposition once the hematoma has been reabsorbed, especially when small diameter stents have been implanted. Use of bare-metal and drug-eluting stents has been reported 5 , 9 but the restenosis rate in this setting is unknown. Since our patient was asymptomatic and spontaneous reabsorption of the hematoma was likely, conservative treatment was decided so as to avoid the need for cardiac surgery or multiple stenting. However, three days later the ST segment again became elevated and stent implantation was required. We decided to implant bare-metal stents instead of drug-eluting stents in view of the positive results reported in the literature and because of the risk of later thrombosis with drug-eluting stents 5,9,10 . IVUS was useful to plan the therapeutic strategy. It differentiated luminal narrowing due to intramural hematoma from a spasm or a diffuse atherosclerotic lesion. It allowed us to assess the extension of the dissection and helped to clarify whether the left main and the proximal LDA artery were also involved, thereby limiting us helping to limit stent implantation. Since coronary angiography provides only a luminal image, stent size calculation is difficult, especially in long dissections. IVUS in this setting allows accurate calculation of vessel size. In coronary dissection there is a risk of inadvertently wiring the false lumen. IVUS assures that the guide-wire is placed in the true lumen, yielding safe stent implantation. In the present case, IVUS helped us to localize the door entrance towards the false lumen and to understand the mechanism of recurrence of angina. Distal LAD artery occlusion was seen at the apical level in the second coronary angiogram. IVUS suggested that the door entrance was a few millimeters beyond the origin of the first diagonal artery and that occlusion of the distal LAD may have been responsible for the was ascribed to sudden re-bleeding into the false lumen on the third day, causing downwards

DISCUSSION

  The pathogenesis of spontaneous coronary artery dissection is related to a coronary artery disruption that causes a false lumen with an intramural hemorrhage. It progressively compresses the true lumen, leading to myocardial ischemia. Diagnosis is usually made by coronary angiography in the setting of an acute coronary syndrome.

  to avoid the need for cardiac surgery or multiple stenting.

Figure 1 .

 1 Figure 1. A: The coronary angiogram showed a diffuse luminal narrowing of the anterior descending coronary artery from close to the first diagonal to the apical segment due to spontaneous coronary dissection (arrows). B: Coronary angiogram after stenting.

Figure 2 .

 2 Figure 2. Above: Angiogram showing occlusion of the apical segment of the anterior descending coronary artery. Numbered cross-sectional intravascular ultrasound images and the corresponding transversal segment in the coronary angiogram of the anterior left descending coronary artery are shown. Below: Four ultrasound images are shown. Slice 1 shows an ultrasound image with a small atherosclerotic plaque in the arterial wall and the origin of the first diagonal branch. No image of dissection is seen. Slice 2. Note the hypodense images suggestive of recent bleeding. Slice 3. Image of an organized hematoma encircling almost the entire circumference of the vessel. Slice 4. Image of an organized hematoma with internal hyperdense areas of unclotted blood.

Figure 1 .

 1 Figure 1. A: The coronary angiogram showed a diffuse luminal narrowing of the anterior descending coronary artery from close to the first diagonal to the apical segment due to spontaneous coronary dissection (arrows).

Figure 2 .

 2 Figure 2. Above: Angiogram showing occlusion of the apical segment of the anterior descending coronary artery. Numbered cross-sectional intravascular ultrasound images and the corresponding transversal segment in the coronary angiogram of the anterior left descending coronary artery are shown. 35x34mm (600 x 600 DPI)

  , A). Diffuse coronary dissection was diagnosed in the LAD artery. The right coronary and circumflex arteries were normal. Creatine phosphokinase was 142 U/L and troponin was T 0.118 µg/L. Conservative treatment (atenolol, nitroglycerine, clopidogrel, acetyl-salicylic acid, clexane and atorvastatine and intermediate doses of enoxaparin) was adopted because the patient was stable.Three days later, she presented a recurrent episode of angina with ST elevation of 2 mm, extending to leads v 5 -v 6. High doses of morphine were needed to alleviate the second coronary angiogram showed persistence of the diffuse narrowing of the distal LAD artery and occlusion at the apical segment (Figure2). Before performing an IVUS, we administered bivalirudin at doses of 0.75 mg/kg/weight followed by a perfusion of 1.5 mg/kg during the procedure. After placing an angioplasty catheter and passing a 0.014" guide-wire to the LAD artery, an IVUS using an Atlantis SR Pro 40 MHz coronary imaging catheter (Boston Scientific, Freemont, Ca) was performed to establish correct positioning of the guide-wire in the true lumen and to assess the extension of dissection. No hematoma was seen in the left main or in the proximal or medium segments of the LAD artery (Figure2, slice 1). A large hypodense image observed about 3 mm below the origin of the diagonal artery was suggestive of an intramural hematoma likely due to recent bleeding (Figure2, slice 2). A dense extraluminal image suggested an organized hematoma extending through the distal segment with internal hypodense areas (Figure2, slice 3-4). Since the patient had angina and ST elevation we decided it was decided to stent the LAD artery. Stenting was started distally, progressing upwards. Six overlapping conventional stents were implanted: Vision 2,25x18 mm (Abbot Vascular, Santa Clara, Ca); Vision 2,75x12 mm; Driver 2,75x18 mm (Medtronic Inc, Minneapolis, MN), Driver 3.0x24 mm, Driver 3x18 mm, Driver 3.25x30 mm. Balloons were inflated to 15-20 atm, including the overlapping stent segments and flow was reestablished (Figure1,B). The chest pain disappeared and the ECG normalized. Creatine phosphokinase was 807 U/L and T troponin was 1.8 µg/L. Echography showed mild hypokinesis with a global left ventricular ejection fraction of 60%. Laboratory tests were performed to rule out associated connective tissue disorders and vasculitis. No etiologic cause of the dissection was found. At discharge, in addition to oral nitroglycerine and atenolol, we prescribed clopidogrel 75 mg/day for one year and acetyl-salicylic acid 100 mg/day indefinitely. A mild apical hypokinesia was observed by echocardiography with a global left ejection fraction of 60%. At 6-month follow-up the myocardial perfusion study using a photon emission computed tomography (SPECT) showed a focal apical necrosis without signs of remaining segments. At 10-18-month follow up the patient was free of ischemic symptoms and a conventional stress test was negative.

  hematoma. This possibility indicates that close surveillance is needed in patients receiving conservative treatment. A multidetector CT angiography or coronary angiogram is recommended to evaluate the healing or progression of the dissection at follow-up.In summary, the present case describes the failure of conservative therapy in a woman with spontaneous coronary dissection, highlighting the relevance of IVUS in diagnosis, and illustrating the good results with stent implantation.

10 LEGENDSFigure 1 .

 101 Figure 1. A: The coronary angiogram showed a diffuse luminal narrowing of the anterior descending coronary artery from close to the first diagonal to the apical segment due to spontaneous coronary dissection (arrows). B: Coronary angiogram after stenting.

Figure 2 .

 2 Figure 2. Above: Angiogram showing occlusion of the apical segment of the anterior descending coronary artery. Numbered cross-sectional intravascular ultrasound images and the corresponding transversal segment in the coronary angiogram of the anterior left descending coronary artery are shown. Below: Four ultrasound images are shown. Slice 1 shows an ultrasound image with a small atherosclerotic plaque in the arterial wall and the origin of the first diagonal branch. No image of dissection is seen. Slice 2. Note the hypodense images in the slice 2 suggestive of recent bleeding. Slice 3. Image of an organized hematoma encircling almost the entire circumference of the vessel. Slice 4. Image of an organized hematoma with internal hyperdense areas of unclotted blood. are seen in the slices 3 and 4.

John Wiley & SonsJournal of Clinical Ultrasound


H Susanto

J Cuevas

P Krüger

Josephson tunneling of dark solitons in a double-well potential

We study the dynamics of matter waves in an effectively one-dimensional Bose-Einstein condensate in a double well potential. We consider in particular the case when one of the double wells confines excited states. Similarly to the known ground state oscillations, the states can tunnel between the wells experiencing the physics known for electrons in a Josephson junction, or be self-trapped. As the existence of dark solitons in a harmonic trap are continuations of such non-ground state excitations, one can view the Josephson-like oscillations as tunnelings of dark solitons. Numerical existence and stability analysis based on the full equation is performed, where it is shown that such tunneling can be stable. Through a numerical path following method, unstable tunneling is also obtained in different parameter regions. A coupled-mode system is derived and compared to the numerical observations. Regions of (in)stability of Josephson tunneling are discussed and highlighted. Finally, we outline an experimental scheme designed to explore such dark soliton dynamics in the laboratory.

Introduction

One fundamental physical phenomenon observable on a macroscopic scale is the Josephson tunneling (JT) of electrons between two superconductors connected by a weak link, predicted by Josephson in 1962 [1]. It is due to the macroscopic wave functions with global phase coherence that have a small spatial overlap. The first observation of this effect was reported by Anderson et al. [2].

Since the only requirement for the occurrence of JT is a weak coupling, other weakly connected macroscopic quantum samples were also expected to admit such tunneling. For neutral superfluids, JT has been observed in liquid 3 He [START_REF] Pereverzev | [END_REF] and 4 He [5]. In the context of Bose-Einstein condensates (BECs) [6,7,8,9,10,11], the prediction was made by Smerzi et al. [12,13,14], followed by the experimental observation where a single [15,16] and an array [17] of short Bose-Josephson junctions (BJJs) were realized. The idea of BJJs has also been extended to a long BJJ [18,19], which mimics long superconducting Josephson junctions. Such a junction can be formed between two parallel quasi onedimensional BECs linked by a weak coupling. Atomic Bose-Josephson vortices (BJVs) akin to Josephson fluxons in superconducting long Josephson junctions [20] have been proposed as well [18,19]. Moreover, it was emphasized that a BJV can transform from and to a dark soliton, due to the presence of a critical coupling at which the two solitonic structures exchange their stability.

The study of JT in BECs considers the tunneling of the Thomas-Fermi cloud, i.e. a continuation of the ground state. The tunneling dynamics has been explained using a two-mode approximation [12,14]. The validity of the approximation has been shown in [34,35]. To improve the applicability regime of such an approximation, modified coupled-mode equations have been presented in, e.g., [30,31,32,33].

It is important to note that in addition to the ground state, nonlinear excitations, such as dark matter waves, can also be created in BECs. Dark soliton dynamics in BECs with single well potentials has been studied theoretically (see a review [START_REF] Konotop | Emergent Nonlinear Phenomena in Bose-Einstein Condensates Theory and Experiment[END_REF]) and experimentally [START_REF] Burger | [END_REF]24,25,27]. Interesting phenomena on the collective behavior of a quantum degenerate bosonic gas, such as soliton oscillations [24,25,26] and frequency shifts due to soliton collisions [27] were observed. The evolution of solitons is of particular interest as the extent to which their behavior can be described in a particle picture is an open question and merits further experimental and theoretical investigation. A combination of soliton physics with the dynamics at weak links within double well potentials will shed light on the collective behavior of excited Bose-Einstein condensates in non-trivial potentials. In this paper, we present an analysis of the dynamics of dark matter waves in a double well potential. Static properties of such a configuration have been recently studied in [28,47]. Here, we show that dark matter waves can also experience stable quantum tunneling between the wells. This implies that localized excitations in higher dimensions, such as vortices, may also experience JT. The (in)stability is obtained using numerical Floquet analysis, which is applied for the first time in the study of JT. The numerical calculations are necessary as the stability of the observed tunneling is not immediately obvious. This is especially the case because dark solitons are higher-order excited states. The possibility that modes with lower energy will be excited is not ruled out by a coupled-mode approximation.

The present paper is outlined as follows. In Section 2, we discuss the governing equation used in the current study. We then solve the equation numerically, where we obtain stable and unstable Josephson tunneling through a numerical path following method. The stability analysis is performed through calculating the Floquet multipliers of the solutions. In Section 3, we derive a coupled-mode approximation describing the tunneling dynamics. Good agreement between the numerics and the approximation is obtained and shown. In Section 4 we present a possible experimental setup to explore the results reported herein. Finally we conclude the work in Section 5.

Josephson tunnelings

Mathematical model

We consider the normalized nonlinear Schrödinger (NLS) equation modelling the BECs (see, e.g., [29] for the scaling)

iψ t + ψ xx + s|ψ| 2 ψ -V (x)ψ = 0, ( 1
)
where ψ is the bosonic field, and t and x is the time and position coordinate, respectively. The parameter s = ±1 characterizes the attractive and repulsive nonlinear interaction, respectively, and V (x) is the external double well potential, which for simplicity is taken as

V = 1 2 Ω 2 (|x| -a) 2 , (2)
with the parameters Ω and a controlling steepness and position of the two minima. The total number of atoms N in the trap is conserved with

N = +∞ -∞ |ψ| 2 dx. (3)
Throughout the present paper, we set s = -1, i.e. we consider repulsive interactions between particles.

For non-interacting particles (s = 0) in a single well potential (a = 0), the governing equation ( 1) can be solved analytically to yield ψ n = e -iEnt φ n (x), where φ n satisfies

φ n+1 = ( Ω 4 √ 2 x - 4 √ 2 Ω ∂ x )φ n , n = 0, 1, 2, . . . , (4)
with

φ 0 = e -Ω 2 √ 2 x 2
, and the chemical potential E n is given by

E n = 1 2 √ 2(2n + 1)Ω.
The excitations φ 0 can be continued to nonzero s, which has been considered in, e.g., [36,37,38,39,40,41,42,43,44,45,46] (see also [48] for discussions on stationary solutions of the NLS equation with a multi-well potential that do not reduce to any of the eigenfunctions of the linear Schrödinger problem). The existence and the stability analysis of continuations of φ n in a double-well potential has been presented in [21], where it was shown that there is a symmetry breaking of the corresponding solutions, i.e. a change of (in)stability from a symmetric to an asymmetric state. One typical manifestation of the instability is a periodic transfer of atoms between the wells, i.e. Josephson tunneling.

As most of Josephson tunneling studied in BECs considers the tunneling of the Thomas-Fermi cloud, which is a continuation of the ground state solution φ 0 , here we consider the tunneling of dark solitons, which can be viewed as continuations of excited states φ n>0 .

Numerical periodic solutions

To look for solutions describing Josephson tunneling, we seek solutions that fulfills the relation ψ(x, T ) = ψ(x, 0), with T being the period of the Josephson oscillations. Such solutions posses double periodicity, i.e. one due to the solitonic nature with a period 2π/E, where E is the chemical potential (intra-well oscillations) and the other one caused by the Josephson effect (inter-well oscillations). Consequently, we can express the solutions in terms of a Fourier series multiplied by a factor related to the stationary character of dark solitons

ψ(x, t) = exp(-iEt) ∞ k=-∞ z k (x) exp(ikωt), (5)
where ω = 2π/T is the Josephson oscillation frequency. These solutions are denoted as commensurate if the commensurability condition E = (q/p)T = (2qπ)/(ωp) is fulfilled, with {q, p} ∈ N. In what follows, we fix p = 1. Commensurate solutions are consequently fixed points of the map ψ(x, 0) → ψ(x, T ) and can be found either by using shooting methods in real space or algebraic methods in Fourier space. In order to do that, we will transform the problem into a discrete one by means of a finite difference discretization with spatial step ∆x = 0.2 and apply the techniques developed for discrete breathers in Klein-Gordon lattices [49,50]. If a shooting method were used, a time step ∆t = 0.02 would be necessary. As the considered oscillations herein have periods about 1500 time units, this method would imply many integration steps. In addition to that, the lack of an analytical Jacobian would also imply the necessity of the numerical determination of this matrix. These facts suggest the suitability of the proposed Fourier space method, which, apart from transforming the set of differential equations into an algebraic one, provides an analytical expression for the Jacobian.

Truncating the Fourier series at k m , i.e. the maximum value of |k|, which has been chosen to be 9 in most of the calculations due to computational reasons, Eq. ( 1) yields a set of nonlinear equations with the k-th component of the dynamical equation set given by

F k (x) ≡ (E -ωk)z k + ∂ 2 x z k -V (x)z k -s km m=-km km n=-km z m z n z k-m+n = 0.(6)
We then obtain the following expression for each component of the Jacobian

∂F k (x) ∂z n (x ) = [E -ωk -V (x)]δ(x -x ) + ∂ 2 xx δ k,n -sδ(x -x ) m z * m z k-n+m + z m (z k-m+n + z * n+m-k ) , (7)
where we have written z k ≡ z k (x) in both equations. Once a periodic solution, say Ψ(x, t), is obtained, to study its (linear) orbital stability one needs to analyze the time evolution of a small perturbation ξ(x, t) to Ψ(x, t). The equation satisfied to leading order by ξ(x, t) is

iξ t + ξ xx -s(2|Ψ| 2 ξ + Ψ 2 ξ * ) -V (x)ξ = 0. ( 8
)
Then, the stability properties of commensurate solutions can be determined by means of a Floquet analysis. It is performed by diagonalizing the monodromy matrix M which is defined as

Re(ξ(x, T )) Im(ξ(x, T )) = M Re(ξ(x, 0)) Im(ξ(x, 0)) . (9)
The linear stability of the solutions requires that the monodromy eigenvalues (also called Floquet multipliers) must be at the unit circle (see, e.g., [49,51,52] for details). In order to get the monodromy with enough accuracy, the simulations must be performed using a time step around ∆t = 0.001. We have calculated commensurate solitons for Ω = 0.1 and a = 10 using the method described above and analysed the stability of those solutions. Presented in the top panels of Figure 1 are two periodic solutions that we obtained together with the time evolution of a dark soliton in a double well potential. The left and right panel respectively corresponds to JT and a transition to macroscopic quantum self-trapping, similarly to the dynamics of the ground state oscillations [12,14].

In the middle panels of Figure 1, we present the distribution of the Floquet multipliers of the two solutions depicted in the top panels in the complex plane. Note that we did not obtain a continuum spectrum of the monodromy due to the discretization of the equations. It is worth noting that as there is a quartet of multipliers that do not lie on the unit circle, one can conclude that the solution in the top right panel is unstable. We show in the bottom panels of Figure 1 a longer time evolution of the solutions in the top panels, where one can see that the solution in the top right panel is indeed unstable. The instability we reported here is a clear evidence that the nonlinearity term in the governing equation ( 1) plays an important role, as all the solutions would have been stable otherwise. A typical instability dynamics is a repulsive interaction between the dark solitons in different wells so that they start to oscillate about the minimum of the wells. This can be clearly observed in the bottom right panel of Figure 1.

We have also obtained periodic solutions for various parameter values. In the top left panel of Figure 2 we show the dependence of the norm (number of atoms) N of tunneling dark solitons when the inter-well oscillation frequency is varied. In the panel, several representative values of q are considered. Note that the possible values of q are not limited to those shown in the graph. As ω is increased further, there is a critical value above which solutions are unstable. Unstable solutions are indicated as dashed line in the top left panel. The solutions can also be continued for decreasing frequencies ω down to a critical value for which the solutions transform into a non-oscillating one (not shown here). In the top right panel of Figure 2 we show the dependence of the growth rate (the logarithm of the maximum modulus of the Floquet multipliers) with respect to ω for q = 47. We also present the growth rate of JT for a fixed ω and q and varying separation distance between the two wells a in the bottom panels of the same figure, i.e. ω = 0.0049 and q = 47. For small a, the solutions tend to a non-oscillating one with one dark soliton in each well, analogously to what occurs for small ω and fixed a.

Coupled-mode approximations and their validity

To describe dark soliton dynamics reported in the previous section, we will readily use a two-mode approximation derived in [31]. Following [31], we write

ψ = √ N (b 2 (t)Φ 2 (x) + b 3 (t)Φ 3 (x)) , Φ 2,3 = Φ + (x) ± Φ -(x) √ 2 , (10)
where Φ ± (x) is a continuation of φ 2,3 (4) for nonzero a satisfying

∂ xx Φ ± + β ± Φ ± -V (x)Φ ± + sN Φ 3 ± = 0, (11)
with the constraint

+∞ -∞ Φ j Φ k dx = δ j,k , i, j = +, -.
Two examples of Φ j , which corresponds to the norm N in the Figure 1 are presented in Figure 3.

Next, for simplicity we write b j (t) = |b j (t)|e iθ j (t) . Equations ( 3) and (10)

imply that |b 2 (t)| 2 + |b 3 (t)| 2 = 1. Defining z(t) = |b 2 (t)| 2 -|b 3 (t)| 2 , ∆θ(t) = θ 3 (t) -θ 2 (t), ( 12
)
one can obtain the equations satisfied by z and ∆θ [31]

dz dt = - ∂H ∂∆θ , d∆θ dt = ∂H ∂z , (13)
where with j, k = +, -. We plot the phase-portrait of (13) in Figure 4 for the two values of N in Figure 1.

H = 1 2 Az 2 -B √ 1 -z 2 cos ∆θ + 1 2 C(1 -z 2 ) cos 2∆θ, (14)
A = 10γ +--γ ++ -γ -- 4 , B = β --β + + γ ++ -γ -- 2 , (15)
C = -2γ +-+ γ ++ + γ -- 4 , γ jk = -sN ∞ -∞ Φ 2 j (x)Φ 2 k (x) dx, (16) (17)
To compare the two-mode approximation with the top panels of Figure 1, we calculate the variable z from the numerics of the full equation ( 1) as [31]

z = 0 -∞ |ψ(x, t)| 2 dx -N/2 NS , S = 0 -∞ Φ + Φ -dx ,
where in the present case S ≈ 0.5. As ∆θ can be calculated immediately, one can compare the numerics and the approximation right away. Shown in Figure 4 are the comparisons, where satisfactory agreement is obtained. As for the instability of the solution in the top right panel, that develops at a later time, it is beyond the validity of any currently available two-mode approximations. One needs a better ansatz for the approximations to capture the stability of the periodic solutions. We conjecture that the invalidity of the approximation is caused by the assumption that the basis functions Φ 2 and Φ 3 are thought to be stable (time-independent), which are not necessarily the case. Note that the validity issue discussed herein is completely different from that in [31]. In [31], the issue is related to the fact that the approximation does not capture the Josephson oscillation of the full equation directly from the beginning, which typically occurs when |sN | 1, while in our case |sN | < 1 and the approximation does capture the existence, but not the stability.

One can observe that the phase portrait in the left panel of Figure 4 has two families of periodic oscillations, i.e. one centred at ∆θ = 0 and the other at ±π. The latter is known as π-oscillations [13]. The stable solution in the top left panel of Figure 2 with q = 50 and the same norm belongs to this family. With this, we predict that all trajectories in the phase portrait are stable.

For the phase portrait in the right panel of Figure 4, one can observe that there are two types of solutions, i.e. Josephson oscillations and running states. The latter type is also referred to as macroscopically quantum self-trapped states. As we have seen that the two-mode approach can provide a good approximation to periodic solutions, one can use the ansatz (10) and direct simulations of (1) to predict whether or not a periodic solution is stable. From several simulations (not shown here), we expect that all trajectories in the phase portrait are unstable.

Based on our analysis above, it is likely that all periodic solutions belonging to the same phase portrait, i.e. having the same norms, will have the same stability property. Nonetheless, one can easily notice in the top left panel of Figure 2 that the critical norms, above which the corresponding solutions are unstable, for different values of ω are close, but not exactly the same. Further analysis whether or not the discrepancy is caused by the finite number of Fourier modes k m is to be addressed in future investigations.

Experimental setup

Dark solitons can be created in a magnetic trap in various ways, including phaseimprinting [START_REF] Burger | [END_REF]24,27], merging of two condensates [25], and passing a penetrable barrier through condensates [53]. In the context of condensate splitting and merging, a method of applying oscillating radio-frequency (rf) fields in combination with static magnetic fields has been recently proven to facilitate good experimental control over creating, tuning, and manipulating double well potentials [54,55,56]. Without changing the static field configuration that provides to very good approximation a parabolic single well trap, for example in atom chip based microtraps [57], modifications of frequency and amplitude of the rf field allow for raising and lowering a splitting potential barrier. The barrier height between the wells is readily controlled in this type of setup. Moreover, inhomogeneities of the rf field, introduced by locally producing the field by an additional conductor on an atom chip [54], can be exploited to introduce an imbalance or slight asymmetry between the two wells. Note that rf field engineering in the context of microtraps with typical distances between trapped atoms and field sources on the order of 10 -100 µm is straightforward as the wavelength of the oscillating field (in the MHz range) is by far sufficiently long to warrant a near-field treatment, with a DC calculation yielding accurate results.

Figure 5 illustrates a possible scheme based on these methods to produce a soliton in an originally harmonically trapped degenerate Bose gas and to observe its subsequent Josephson oscillations in a double well potential. The protocol starts with a simple harmonic magnetic trap, as can be produced on an atom chip by a Z-shaped wire [57]. The gas is then split with rf fields into a symmetric double well. The double well is then slightly imbalanced so that after a time of typically a few ms, a relative phase difference of the two clouds of φ = π will have accumulated. Merging the potentials will now result in a solitonic excitation of the combined gas (see, e.g., [58]). Slight controllable deviations ∆φ from π will produce a (slowly) oscillating soliton. After raising the barrier once again, the tunneling dynamics of the soliton can now be studied. The parameters for this protocol can be chosen similarly to those used in previous experiments in double well potentials [54,56], i.e. condensates of ∼ 10 4 -10 5 atoms of for example 87 Rb can be loaded in a standard magnetic microtrap with oscillation frequencies of ω ⊥ ∼ 2π × 1 kHz in the transverse direction of strong confinement. Typical separations of the two wells in the double well potential will be on the order of a few microns. The scheme can be extended to multiple solitons by splitting the potential in more than two wells, for example by applying multiple frequency component rf fields [59], relevant in the study of interactions of different numbers of dark solitons in each well.

Conclusion and future work

We have studied dark soliton dynamics in a double well potential, where it has been shown that dark solitons can experience JTs between the wells. A coupled-mode approximation has been derived to explain the observations. Numerical stability analysis based on the full governing equation has been performed to show that JTs can be stable. Through path following methods, unstable solutions were also obtained. An experimental scheme designed to explore such dark soliton dynamics in the laboratory also has been outlined.

A natural problem to follow the tunneling reported above is when two different numbers of dark solitons are loaded into each of the minima. We present in Figure 6 interactions between one and two dark solitons. One can observe that there is an interference pattern analogous to the acoustic beating pattern in the interaction of two continuous waves with slightly different frequencies. As shown in the left panel indicated by the dashed ellipses, there are two levels of modulated patterns; the big ellipse shows one tunneling period modulated by the oscillation in the small ellipse. In the right panel, we zoom in on the small ellipse to show that a beating pattern also occurs on a smaller scale. It can be calculated that the oscillation period in the small ellipse when a 1 is approximately T ≈ 2π/Ω, which is in accordance with the numerical result. A multi-mode approximation can be obtained as before as briefly discussed in [31]. It is then interesting to study the stability of such interactions. Together with the question on a better ansatz that can predict the stability of periodic solutions reported herein, this is currently under investigation and will be reported elsewhere.

Figure 1 .

 1 Figure 1. (Top) The first few oscillations of the atom density |ψ(x, t)| 2 for dark solitons in a double well potential with Ω = 0.1, a = 10, and (left) ω = 0.00450 and (right) ω = 0.00520, which respectively corresponds to N = 0.0340 and N = 0.7677. In both cases, the initial conditions are obtained from a numerical continuation with q = 47 (see the text). (Middle) Floquet multiplier distributions corresponding to solutions in the top left and right panel, respectively. (Bottom) Longer time evolutions of the top panels where one can see that the solution in the top right panel is indeed unstable.

Figure 2 .

 2 Figure 2. The top left panel presents the dependence of the norm with respect to ω for dark solitons with a = 10. Dashed lines indicate unstable solutions.Here, q sweeps the values between 40 and 50. The top right panel shows the dependence of the growth rate with respect to ω for q = 47. Bottom panels depict the norm and the growth mode of tunneling dark solitons with fixed ω = 0.0049 and q = 47 for varying a.

Figure 3 .

 3 Figure 3. The second and third collective modes of the confining potential V (x) (dash-dotted) for (solid) N = 0.034 and (dashed) N = 0.7677, with s = -1.

Figure 4 .

 4 Figure 4. The phase-portraits of (13) for the two values of N in Figure 1, i.e. (left) N = 0.0340 and (right) N = 0.7677. Thick symbols correspond to the periodic solutions shown in Figure 1.

Figure 5 .

 5 Figure 5. Schematic implementation of a protocol for controlled production of a soliton in a double well potential (see text).

Figure 6 .

 6 Figure 6. (Color online) Interactions between two different excitations when loaded into each of the minima of the magnetic trap. Big and small dashed ellipse indicates one tunneling period of the excitation in the left and right well, respectively. The bottom panel zooms in the small dashed ellipse. The total number of atoms is small, similar to that in Figure 1.

Acknowledgments

We acknowledge fruitful discussions with Andrea Trombettoni and Panos Kevrekidis. JC acknowledges financial support from the MICINN project FIS2008-04848.


A. L'Huillier J M Dahlström

J Mauritsson

Quantum mechanical approach to probing the birth of attosecond pulses using a two-color field

Keywords: numbers: 32.80.Rm, 32.80.Qk, 42.65.Ky

We investigate the generation of even and odd harmonics using an intense laser and a weak second harmonic field. Our theoretical approach is based on solving the saddle-point equations within the Strong Field Approximation. The phase of the even harmonic oscillation as a function of the delay between fundamental and second harmonic field is calculated and its variation with energy is found to be in good agreement with recent experimental results. We also find that the relationship between this phase variation and the group delay of the attosecond pulses, depends on the intensity and wavelength of the fundamental field as well as the ionization potential of the atom.

Introduction

Attosecond pulses [START_REF] Paul | Observation of a train of tttosecond pulses from high harmonic generation[END_REF][START_REF] Hentschel | Attosecond metrology[END_REF] are created through the interaction between intense infrared (IR) laser fields and atoms or molecules in a process known as high-order harmonic generation (HHG). This process is initiated by the creation of an electron wave packet through tunneling ionization, followed by acceleration in the laser field and recombination with the ion core resulting in extreme ultraviolet (XUV) emission confined to a fraction of the laser cycle. The generation process can be controlled by shaping the driving laser field, e.g. by coherently adding laser fields of different wavelengths to the fundamental laser field [START_REF] Long | Model calculations of polarization-dependent two-color high-harmonic generation[END_REF][START_REF] Gaarde | Theory of high-order sum and difference frequency mixing in a strong bichromatic laser field[END_REF][START_REF] Radnor | Carrier-wave steepened pulses and gradient-gated high-order harmonic generation[END_REF][START_REF] Chipperfield | Ideal waveform to generate the maximum possible electron recollision energy for any given oscillation period[END_REF]. Most investigations have been carried out by adding the second harmonic generated in a doubling crystal [START_REF] Figueira De Morisson Faria | Phase-dependent effects in bichromatic high-order harmonic generation[END_REF][START_REF] Mauritsson | Attosecond Pulse Trains Generated Using Two Color Laser Fields[END_REF][START_REF] Mansten | Spectral shaping of attosecond pulses using two-colour laser fields[END_REF][START_REF] Ishii | Quantum path selection in high-harmonic generation by a phase-locked two-color field[END_REF][START_REF] Dudovich | Subcycle spatial mapping of recollision dynamics[END_REF][START_REF] Frolov | Analytic description of high-order harmonic generation by atoms in a two-color laser field[END_REF][START_REF] He | Interference effects in two-color high-order harmonic generation[END_REF][START_REF] Eilanlou | Frequency modulation of high-order harmonic fields with synthesis of two-color laser fields[END_REF], with various goals ranging from optimization of the conversion efficiency, characterization of the emitted attosecond pulses and production of attosecond pulse trains with one pulse per cycle of the fundamental field. Several parameters can be varied in these experiments: the intensity ratio, the phase difference and the relative polarization direction.

When the second harmonic field is a weak perturbation to the fundamental, the generation of odd harmonics is barely changed, but the induced symmetry breaking leads to the appearance of weak even harmonics. High-order harmonic spectra recorded as a function of delay between the two fields show that the intensity of the even harmonics is modulated and that this modulation has an offset depending on the harmonic order, as shown in Figure 1(a) [START_REF] He | Interference effects in two-color high-order harmonic generation[END_REF]. These photon spectrograms are at a first glance similar to the photoelectron spectrograms of the RABITT (Reconstruction of Attosecond Bursts by Interference of Two-photon Transitions) method [START_REF] Paul | Observation of a train of tttosecond pulses from high harmonic generation[END_REF][START_REF] Mairesse | Attosecond synchronization of high-harmonic soft X-rays[END_REF] used to characterize attosecond pulses. Dudovich et al. [START_REF] Dudovich | Measuring and controlling the birth of attosecond XUV pulses[END_REF] suggested that the two-color HHG spectrograms could be used to determine the emission times of the attosecond pulses in situ, thus "probing the birth of attosecond pulses". This method was then applied by Doumy et. al. [START_REF] Doumy | Attosecond Synchronization of High-Order Harmonics from Midinfrared Drivers[END_REF], using laser systems of different wavelengths, and by us [START_REF] Dahlström | Atomic-and macroscopic measurements of attosecond pulse trains[END_REF], in a direct experimental comparison with the RABITT method.

In this article we investigate HHG by a laser field and its second harmonic by solving the Schrödinger equation within the Strong Field Approximation (SFA), using the saddle point equation method [START_REF] Lewenstein | Rings in above-threshold ionization: A quasiclassical analysis[END_REF]. We assume throughout that the second harmonic is weak and can be considered as a small perturbation to the fundamental field. We calculate the phase variation of the even harmonic oscillation and compare it with the harmonic emission times. Our model allows us also to interpret the rapid variation of this phase at high energy, observed in several experiments [START_REF] Dahlström | Atomic-and macroscopic measurements of attosecond pulse trains[END_REF][START_REF] He | Interference effects in two-color high-order harmonic generation[END_REF], as a change of the dominant quasiclassical trajectory from the short to the long one [Figure 1(b)]. The article is organized as follows: In section 2 we review and compare classical and quasiclassical electron trajectories for one-color (Sec. 2.1) and two-color HHG (Sec. 2.2). In section 3 we study the generation of even harmonics close to the cutoff (Sec. 3.1) and we relate the phase of the even harmonic oscillation to the emission times of the attosecond pulses (Sec. 3.2). In section 4 we summarize our results by presenting a more general relationship between the phase of the even harmonic oscillation and the [START_REF] He | Interference effects in two-color high-order harmonic generation[END_REF]. The diamonds ( ) indicate the experimental delays corresponding to maxima of the modulation in the even harmonic orders. (b) Calculated delays corresponding to maxima of the even orders for an effective intensity of 1.4 × 10 14 W/cm 2 . Full (dashed) curve are calculated for the short (long) branch labeled 1 (2). Below (above) harmonic order 28 the short (long) branch fits the experimental data ( ). emission times.

Quasiclassical trajectories

Our method is based on the stationary phase method (also referred to as the saddle point method) to solve the Schrödinger equation for an atom exposed to a strong laser field using the SFA [START_REF] Lewenstein | Theory of high-order harmonic generation by low-frequency laser fields[END_REF]. In particular, we consider the interaction of an atom with a strong IR laser field (frequency ω) and a weak second harmonic (frequency 2ω). In the SFA, the electron will (I) tunnel into the continuum, (II) accelerate in the strong laser field and then (III) return to the atom and emit a harmonic photon (frequency Ω). The phase of the harmonic radiation is related to the quasiclassical action of the electron

S( p, t, t 0 ) = t t 0 dt   ( p -e A(t )) 2 2m + I p   , (1)
where A(t) is the vector potential of the laser field, I p is the ionization potential of the atom; and p, t 0 , t, m and e are the drift momentum, tunneling time, return time, mass and charge of the electron respectively. The high-order harmonic emission will mainly originate from the stationary points of S -hΩt, with respect to all variables [ p, t, t 0 ], which satisfy the following three equations [START_REF] Lewenstein | Theory of high-order harmonic generation by low-frequency laser fields[END_REF] t

t 0 dt e A(t ) = (t -t 0 ) p, (2)
( p -e A(t 0 )) 2 2m = -I p , (3)
( p -e A(t)) 2 2m = hΩ -I p . ( 4
)
The electron is thus required to return to the atom at time t [Eq. 2], to undergo complex tunneling at time t 0 [Eq. 3] and to satisfy energy conservation [Eq. 4]. For any realistic atom, we have I p > 0 which implies that the electron must tunnel into the continuum. The tunneling process results in damping of the electron trajectories, i.e. complex stationary points in the harmonic plateau [START_REF] Becker | Above-threshold ionization: from classical features to quantum effects[END_REF]. The electron trajectories beyond the cutoff are always strongly damped (also for I p = 0), because they are always classically forbidden, corresponding to large imaginary components of the stationary points. The physical emission time of a given harmonic is given by the real part of the complex emission time [START_REF] Mairesse | Attosecond synchronization of high-harmonic soft X-rays[END_REF][START_REF] Chirilȃ | Emission times in high-order harmonic generation[END_REF][START_REF] Varju | Frequency chirp of harmonic and attosecond pulses[END_REF].

One-color case

We first consider the one-color case with a vector potential, A(t) = A 1 sin(ωt), where ω is the angular frequency of the fundamental laser light. The complex stationary points [ p (n) , t (n) , t

0 ] are calculated as a function of the high-order harmonic photon angular frequency, Ω. The index n is used to separate different sets of solutions, where n = 1 corresponds to the short branch; and n = 2 corresponds to the long branch. A direct comparison between the stationary points for I p = 15.76 eV (Ar) and I p = 0 eV (referred to as the classical case) is shown in Figure 2. The detailed behavior of these stationary points will prove to be important not only for the generation of attosecond pulses from one-color HHG, but also in the quantitative analysis of two-color HHG. The classical case corresponds to solving Eqs. 2-4 for I p = 0. The stationary points are first real, describing classical trajectories in the continuum. Using classical trajectories to explain the HHG process leads to the approximate cutoff law: hΩ max ≈ 3.2U p + I p [START_REF] Krause | High-order harmonic generation from atoms and ions in the high intensity regime[END_REF][START_REF] Corkum | Plasma perspective on strong-field multiphoton ionization[END_REF]. Harmonics above this cutoff can not be generated because there are no electrons returning with suffiecient kinetic energy. Quantum mechanically, we see that the stationary points quickly develop large imaginary components above the cutoff, which result in an exponential damping of the harmonic yield. Solving Eqs. 2-4 for I p = 0, the exact cutoff law is found to be hΩ max = 1.3I p + 3.2U p [START_REF] Lewenstein | Theory of high-order harmonic generation by low-frequency laser fields[END_REF]. In Figure 2 we have, therefore, shifted the classical solutions by 1.3I p so that the cutoff harmonic of the classical case coincides with the quantum mechanical case of argon for a more meaningful comparison. The similarities and differences between the classical case (red line) and the quantum case (blue line) are clearly observed: The real parts of the drift momenta [Fig. 2(a)] are almost the same in the two cases, apart from a small systematic shift. The real parts of the return times [Fig. 2(c)] overlap almost perfectly in the main part of the plateau for both the short (line) and long (dashed) branches. This demonstrates the usefulness of the classical model for understanding the intrinsic chirp of the returning electron wave packet and the resulting chirp of the attosecond light pulses. The real parts of the tunneling times [Fig. 2(e)] are, however, quite different. In the lower part of the plateau there is a discrepancy of a factor of two for the short = 0 eV (thin red). The classical case is shifted by 1.3I p to match the quantum mechanical cutoff. The short branch is a line; while the long branch is a dashed line. The intensity is 2 × 10 14 W/cm 2 and the wavelength is 800 nm.

branch. The short branch trajectories tunnel at earlier times compared to the classical case due to the ionization potential. This implies longer times in the continuum and an increased amount of phase acquired, as the path of integration extends in Equation 1. For the long branch the trend is opposite and smaller. The imaginary parts of the stationary points [Fig. 2(b,d,f)] lead to exponential damping of the electron amplitudes. This is especially clear for the imaginary part of the tunneling time [Fig. 2(f)] where the complex part is zero in the classical case where no tunneling is required; while, in the complex case, the short branch will suffer more damping than the long branch due to a lower instantaneous electric field strength at the time of tunneling.

The high-order harmonic dipole from one half period of the laser field can be approximated as a sum of stationary contributions

x Ω = d 3 p T /2 0 dt t -∞ dt 0 Λ exp [iS( p, t, t 0 )/h -iΩt] ≈ n=1,2 x (n) Ω . (5)
These discrete contributions correspond to different sets of stationary phase solutions

x (n) Ω = Λ (n)   ih 5 det[M (n) ij ]   1/2 exp iS (n) /h -iΩt (n) , (6)
where

S (n) = S( p (n) , t (n) , t (n) 0
) and where

M ij = ∂ 2 ∂i∂j [S( p, t, t 0 ) -hΩt] , (7)
is the Hessian matrix of second-order derivatives of the Legendre transformed action with i and j being any of our variables [ p, t, t 0 ]. The Hessian matrix is complex symmetric and we have assumed that it can be diagonalized so that the one-dimensional stationary phase approximation can be applied to all five integrals independently. The prefactor Λ = d( p -e A(t)) * E(t 0 )d( p -e A(t 0 )), describes the dipole transitions to the continuum at time t 0 , and the subsequent recombination to the ground state at time t. In the following, we will assume that Λ is slowly varying and that it, therefore, does not affect the stationary points. A more exact analysis could, however, also include the effect of the energy dependent atomic scattering phases, η l , in the stationary phase equations (Eqs. 2-4) (from the recombination matrix element d( p -e A(t)) * ∝ exp[-iη l ]). This would lead to a small change in the stationary points, which would be especially interesting to study in argon due to an unusually strong variation of the scattering phase in the present energy region [START_REF] Kennedy | Photoionization of the Noble Gases: Cross Sections and Angular Distributions[END_REF].

Assuming the process to be periodic, the total dipole response, X Ω , is found by summing the stationary points from one whole period of the fundamental laser field, 0 < t < T , i.e. two adjacent half periods. There are thus two contributions from each branch,

x (n) Ω (t + T /2, t 0 + T /2) = -x (n) Ω (t, t 0 ) exp [-iΩT /2] , (8)
separated by a half period, T /2. The overall minus sign comes from the change of sign of E(t); while the phase factor originates from the Fourier component in the Legendre transformed action. The total dipole response becomes

X Ω ≈ n x (n) Ω {1 -exp [-iΩT /2]} = 2 n x (n) Ω × 1, Ω/ω is odd 0, Ω/ω is even , (9)
where the two contributions add constructively for odd orders; and destructively cancel for even orders.

Two-color case

We now consider a two-color laser field composed of a fundamental laser field and a weak second harmonic with the same polarization,

A(t) = A 1 sin(ωt) + λ A 2 sin(2ωt + φ),
where λ is a small perturbation parameter. It is possible to solve the two-color highorder harmonic emission using the stationary phase equations [Eqs. [START_REF] Hentschel | Attosecond metrology[END_REF][START_REF] Long | Model calculations of polarization-dependent two-color high-harmonic generation[END_REF][START_REF] Gaarde | Theory of high-order sum and difference frequency mixing in a strong bichromatic laser field[END_REF] directly, but this requires evaluation of the system at all values of φ. We will follow a different route where the second harmonic is treated as a perturbation and only the stationary points of one-color HHG need to be calculated. The two-color action is expanded in λ as

S ≈ t t 0 dt   [ p -e A 1 (t )] 2 2m + I p   S 1 -λ t t 0 dt [ p -e A 1 (t )][e A 2 (t , φ)] m λσ . (10)
The first term in Eq. 10 corresponds to the action in the one-color case, S 1 ; and the second term is the correction term due to the interaction with the second harmonic field, σ = σ(φ). The correction term can be calculated as

σ = e m p A 2 cos(2ωt + φ) 2ω + e A 1 A 2 sin(3ωt + φ) 6ω - sin(ωt + φ) 2ω t t 0 , (11)
where σ depends on the ionization potential through the stationary points. Note that the derivation of Eq. 11 using the quantum mechanical stationary points differs from those presented previously [START_REF] Dudovich | Measuring and controlling the birth of attosecond XUV pulses[END_REF][START_REF] Dahlström | Atomic-and macroscopic measurements of attosecond pulse trains[END_REF], because it includes the effect of the ionization potential within the SFA.

The two-color high-order harmonic dipole can be calculated by using Eq. ( 6) and by adding an additional slow factor due to the second harmonic. In analogy with Eq. ( 9), the dipole from one period of the fundamental field contains one discrete contribution for each branch and half period,

X Ω ≈ n x (n) Ω exp iσ (n) -exp -iσ (n) - iΩT 2h = 2 n x (n) Ω ×    cos σ (n) /h , Ω/ω is odd i sin σ (n) /h , Ω/ω is even (12) where x (n)
Ω is the half period contribution in the one-color field, and where the following property

σ (n) (t + T /2, t 0 + T /2) = -σ (n) (t, t 0 ) ( 1 3 )
relating the two-color phase between adjacent half periods for a given branch n is used. The intensity of the even order harmonic emission from a single atom can be approximated by

I Ω ∝ X Ω 2 ≈ 2 n x (n) Ω iσ (n) /h 2 , (14)
which is valid for |σ (n) | π. We define the intensity from a specific branch, n, as

I (n) Ω (φ) ∝ σ (n) (φ) 2 . ( 15
)
This (artificial) separation of the branches can be realized in macroscopic medium either by phase matching in a long gas cell [START_REF] He | Interference effects in two-color high-order harmonic generation[END_REF] or by spatial separation in the far field [START_REF] Dudovich | Subcycle spatial mapping of recollision dynamics[END_REF][START_REF] Dudovich | Measuring and controlling the birth of attosecond XUV pulses[END_REF].

Subcycle delay dependence of even harmonics

One fascinating aspect of two-color HHG is that it depends on the subcycle delay (or relative phase φ) between the laser fields. Dudovich et al. [START_REF] Dudovich | Measuring and controlling the birth of attosecond XUV pulses[END_REF] have proposed to use the φ-dependence of even-order harmonics in order to estimate the emission times of the attosecond pulses in situ. We will refer to the relative phases that maximize the intensity of the even harmonics as the in situ phases: φ intensity in (a,b) is realistic for typical laser pulses of 30 fs in argon; while the intensity in (d,e) is greater than the saturation intensity and, therefore, not experimentally feasible for multicycle laser pulses. The aim of our analysis is to distinguish between what can be measured experimentally, φ

Behavior close to cutoff

The similarity between the in situ phases and the emission times is striking, especially for the high intensity [Fig. 3 (d,e,f)] where the short and long branch merge in the cutoff at harmonic 63. At lower intensity [Fig. 3 (a)] the in situ phases do not merge in the cutoff. Intuitively, one might think that the short and long branch should merge in the cutoff, but this is not necessary in the quantum mechanical case. The stationary points of the short and long branch do not merge on the imaginary axis in the cutoff [Fig. 2 (b,d,f)]. The different behavior of the in situ phases for the short and long branch in the cutoff is, hence, an amplitude effect rather than a pure phase effect. This is verified by inserting only the real part of the stationary points into Eq. ( 15) which does indeed yield coincidental cutoff behavior of both branches, as expected for pure phase effects. Furthermore, it is the long branch that remains physical beyond the cutoff; while the strange behavior of the short branch arises from a set of stationary points that become unphysical beyond the cutoff [START_REF] Lewenstein | Phase of the atomic polarization in high-order harmonic generation[END_REF]. In the classical model [Fig. 3 (c,f)], the in situ phases always merge in the cutoff since there are no amplitude effects (damping) in the plateau [Fig. 2 (b,d,f)].

It is also worth to note that the in situ phases of the two branches intersect at lower harmonic orders than the cutoff. This "intra-plateau crossing" is marked with a cross (×) in Figure 3 (a,d) and it should not be confused as the position of the cutoff. At high intensities it is easy to distinguish between the intra-plateau crossing and the cutoff; while at low intensities they may be separated by a few harmonic orders only.

In Figure 1 we present an experimental result where φ 0 is linear from harmonic 22 to harmonic 28. At higher harmonic orders, a dramatic bend is observed [START_REF] He | Interference effects in two-color high-order harmonic generation[END_REF][START_REF] Dahlström | Atomic-and macroscopic measurements of attosecond pulse trains[END_REF]. Using our quantum mechanical model we identify the lower orders as part of the short branch, n = 1, (full curve); while the higher-orders are identified as the long branch, n = 2, (dashed curve). We stress that the dominance of the long branch close to the cutoff is a new result which appears already at the single atom level when the short branch becomes unphysical. This new finding illustrates the usefulness of quantitative probing of HHG using a perturbative field. This effect would be very difficult to observe with the RABITT method because the corresponding emission times always merge in the cutoff, see Fig. 3 (b,e). The classical model fails to reproduce the bend [Fig. 3 (c)] because the corresponding intra-plateau crossing occurs at much lower harmonic orders (not shown). Having discussed the details of the behavior close to the cutoff, we now turn our attention to the central part of the harmonic plateau.

Ratio of in situ phases and emission times

In the following, we consider the first spectral derivative of the in situ phase, ∂φ (n) 0 /∂Ω, which we wish to compare with the group delay dispersion (GDD), ∂t (n) R /∂Ω. Both quantities are evaluated in the central 50% of the harmonic plateau, i.e. in a region set by 1.3I p + 3.2U p ± 0.25 × 3.2U p , where both φ

(n) 0 (Ω) and t (n)
R are linear to a very good approximation. We avoid fast and nonlinear variations both close to the ionization potential (hΩ = I p ) and close to the cutoff (hΩ = 1.3I p +3.2U p ) using this central region. We define the ratio between the two quantities as

γ (n) = -ω ∂t (n) R ∂Ω / ∂φ (n) 0 ∂Ω = -ω ∂t (n) R ∂φ (n) 0 . (16)
The GDD can in principle be obtained from the φ-dependence of the even-order harmonics as ∂t

(n) R /∂ω = -γ (n) ∂φ (n)
0 /∂ω/ω. This relation would be very useful if γ (n) was a constant (or at least a constant for each branch n). Unfortunately we show that γ (n) depends on the ionization potential of the atom, as well as the laser intensity and wavelength.

Role of ionization potential

In this subsection we study how γ (n) , varies with laser intensity for a given laser wavelength (800 nm) and different atomic species (helium, argon, sodium and the classical case). In our model of γ (n) , atom specific properties enter only through the value of the ionization potential, I p . In Figure 4, we plot γ (n) for the short branch (a) and the long branch (b). In the classical case (I p = 0) we find constant ratios: γ (1) ≈ 1.1 for the short branch and γ (2) ≈ 0.84 for the long branch; while in all realistic cases, where I p > 0, there is a significant change in γ (n) as a function of the fundamental intensity. In the case of argon there is an especially strong variation of the ratio: from 1.6 to 0.5 for the short branch over the given intensity range. This strong intensity dependence is problematic for the experimental determination of twocolor HHG since the exact effective intensity is often unknown in experiments. The variation of the ratios is smaller for the long branch, because the stationary points are more similar to the corresponding classical case, as seen in Figure 2. The variation of the ratios is smaller for sodium than for helium because of the smaller ionization potential. Sodium is chosen as an example to show that there is a significant difference between the classical and quantum mechanical cases even for a relatively small ionization potential. To use the two-color HHG approach for characterization of attosecond pulses would require an accurate determination of the laser intensity, as well as a quantum mechanical calculation of γ (n) .

Intensity [W/cm 2 ] (b) (b) (b) 0.5 1 1.5 2 γ (1)
Next we comment on our previous work [START_REF] Dahlström | Atomic-and macroscopic measurements of attosecond pulse trains[END_REF] where we compared the in situ method and the RABITT method for GDD of attosecond pulses. An explanation for the good agreement we found between the two methods may come from the crossing of γ (1) for argon occurring at 1.8 × 10 14 W/cm 2 with the classical limit, see Fig. 4(a). It would be interesting to see more experimental results, carried out on different atoms.

Role of laser wavelength

Finally, we study the dependence of the ratios, γ (n) , with laser wavelength. The ratios are calculated for 800 nm, 1.3 µm and 2 µm, corresponding to a Titanium-Sapphire laser system and two mid IR laser sources. We use the ionization potential of argon. Similar to Fig. 4, all ratios look different when they are plotted as a function of laser intensity, see Figure 5 (a) for the short branch and (b) for the long branch. The ratios of the short branch show larger variations than those of the long branch. Furthermore, it is seen that the longer wavelengths lead to less variation of the ratios over intensity. The recent experiments of Doumy et. al. [START_REF] Doumy | Attosecond Synchronization of High-Order Harmonics from Midinfrared Drivers[END_REF] were carried out at these wavelengths. Applying our method improves the scaling law for the GDD from λ -0.77 to λ -1.05 , which is closer to the expected λ -1 scaling of the harmonic chirp times intensity. We stress that in order to use the calculated ratios to determine the GDD, the experimental intensity must first be measured as accurately as possible independently of the two-color HHG scheme.

Intensity [W/cm 2 ] (b) (b) (b) 0.5 1 1.5 2 γ (1)

Discussion and conclusions

In our quantum mechanical derivation of two-color HHG we find that both amplitude and phase effects are important. The ratio of the ponderomotive energy and the ionization potential, U p /I p , serves as a measure on how "classical" or how "quantum" the electron trajectories are. Choosing this ratio as our x-axis for the data in Fig. ( 4)-( 5), we find that all individual ratios γ (n) , follow a universal curve, as shown in Figure 6. Photoelectron emission ranging from the photon picture to the tunneling picture is described in the theory of Keldysh [START_REF] Keldysh | Ionization in the Field of a Strong Electromagnetic Wave[END_REF] where the ratio I p /2U p 1 (corresponding to U p /I p 1) implies efficient tunnel ionization. In this limit we find that γ (n) slowly converges toward the classical limit, I p = 0. One should look at Fig. 6 with some caution since the SFA is derived for the long wavelength limit requiring: I p hω and U p /I p > 1/2. It is clear that the behavior of the HHG process changes dramatically around U p /I p ≈ 1, i.e. when the kinetic energy of the electron is close to the potential energy of the atom. In conclusion, we have studied the HHG process perturbed by a weak second harmonic field within the SFA. We find that the dependence of the even harmonics on the subcycle delay between the two fields, can not be understood using classical theory.

Our calculations show good agreement with experimental results [START_REF] He | Interference effects in two-color high-order harmonic generation[END_REF][START_REF] Dahlström | Atomic-and macroscopic measurements of attosecond pulse trains[END_REF] showing the change of behavior at high energy, explained as a change of dominant quasiclassical branch of trajectories. We stress that there is an intra-plateau crossing between the short and the long branch which does not coincide with the true cutoff. Furthermore, we calculate the ratio between the GDD of the attosecond pulses and the phase variation of the even-order harmonics, γ (n) , as a function of intensity, wavelength and ionization potential. The analysis method called in situ probing of the birth of attosecond pulses [START_REF] Dudovich | Measuring and controlling the birth of attosecond XUV pulses[END_REF] must be improved by considering the influence of the atomic properties and laser parameters, before it can be applied for quantitative experimental studies. Using the classical analysis will only lead to a qualitative prediction for the GDD of the attosecond pulses with the correct sign. It will be interesting to compare the results of our quantum mechanical approach to probing the birth of attosecond pulses using a two-color field with refined calculations.

Figure 1 .

 1 Figure 1.(a) Experimental two-color high-order harmonic spectrogram over relative delay[START_REF] He | Interference effects in two-color high-order harmonic generation[END_REF]. The diamonds ( ) indicate the experimental delays corresponding to maxima of the modulation in the even harmonic orders. (b) Calculated delays corresponding to maxima of the even orders for an effective intensity of 1.4 × 10 14 W/cm 2 . Full (dashed) curve are calculated for the short (long) branch labeled 1 (2). Below (above) harmonic order 28 the short (long) branch fits the experimental data ( ).

Figure 2 .

 2 Figure 2. Real and imaginary parts of the stationary points, [p, t, t 0 ]. Ionization potentials: argon I (Ar) p = 15.76 eV (thick blue) and classical I (0) p

Figure 3 .

 3 Figure 3. Quantum mechanical in situ phases, φ (n) 0 , between the fundamental and weak second harmonic field that maximize the even harmonic emission in argon (I (Ar) p = 15.75 eV) for intensities of (a) 1.5 × 10 14 W/cm 2 and (d) I = 4 × 10 14 W/cm 2 . The in situ phases are presented as delays, τ (n) 0 = -φ (n) 0 /2ω in units of T/2. The corresponding high-order harmonic emission times are shown in (b) and (e), in units of T . Classical in situ phases (I (0)p = 0) for intensities of (c) 1.5×10 14 W/cm 2 and (f) I = 4 × 10 14 W/cm 2 . The short (long) branch is a full (dashed) curve. T = 2π/ω is the period of the fundamental laser field.

  ), and the desired emission times, t (n) R (Ω). The in situ phases from the classical model (I p = 0) are plotted in Figure 3 (c) and (f) for comparison.

Figure 4 .

 4 Figure 4. Ratios for different ionization potentials as a function of fundamental intensity for (a) the short branch and (b) the long branch. Symbols are helium I (He) p = 24.58 eV, 2 argon I (Ar) p = 15.76 eV, sodium I (Na) p = 5.14 eV and • classical I (0) p = 0 eV.

Figure 5 .

 5 Figure 5. Ratios for different laser wavelengths as a function of fundamental intensity for (a) the short branch and (b) the long branch. Symbols are: λ = 800 nm, 2 λ = 1.3 µm, λ = 2 µm are for argon I (Ar) p = 15.6 eV; while symbol • is the classical case I (0) p = 0 eV, which is independent of λ.

Figure 6 .

 6 Figure 6. Universal ratio between the group delay dispersion and the derivative of the in situ phases as a function of the ratio of ponderomotive energy and ionization potential for (a) the short branch and (b) the long branch. The classical limit, I (0) p = 0, is reached very slowly see gray arrows.

Acknowledgments

The authors would like to thank K. Varjú and K.J. Schafer for helpful discussions.