Léo Terziman

Maud Marchal

Mathieu Emily

Franck Multon

Anatole Lécuyer

Shake-Your-Head: Revisiting Walking-In-Place for Desktop Virtual Reality

Keywords: H.5.2 [Information Interfaces and Presentation]: User Interfaces-Input Devices and Strategies, Interaction Styles, User-centered Design; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism-Virtual Reality Walking, Walking-In-Place, Camera Motions, Head Tracking, First-Person-Navigation, Desktop Virtual Reality

The Walking-In-Place interaction technique was introduced to navigate infinitely in 3D virtual worlds by walking in place in the real world. The technique has been initially developed for users standing in immersive setups and was built upon sophisticated visual displays and tracking equipments.

In this paper, we propose to revisit the whole pipeline of the Walking-In-Place technique to match a larger set of configurations and apply it notably to the context of desktop Virtual Reality. With our novel "Shake-Your-Head" technique, the user is left with the possibility to sit down, and to use small screens and standard input devices such as a basic webcam for tracking. The locomotion simulation can compute various motions such as turning, jumping and crawling, using as sole input the head movements of the user. We also introduce the use of additional visual feedback based on camera motions to enhance the walking sensations.

An experiment was conducted to compare our technique with classical input devices used for navigating in desktop VR. Interestingly, the results showed that our technique could even allow faster navigations when sitting, after a short learning. Our technique was also perceived as more fun and increasing presence, and was generally more appreciated for VR navigation.

Introduction

Navigation is one of the fundamental tasks needed for 3D interaction with Virtual Environments (VE) [START_REF] Bowman | 3D User Interfaces: Theory and Practice[END_REF]]. The possibility to walk inside the VE is necessary in many applications of Virtual Reality (VR) such as for virtual visits (review of architectural and urban projects) or virtual training (technical procedures, military infantry).

In order to provide VR users with realistic sensations of walking while keeping them in their limited workspace, numerous types of VR interfaces have been proposed so far [START_REF] Bowman | 3D User Interfaces: Theory and Practice[END_REF]. Locomotion interfaces, such as unidirectional or omnidirectional treadmills [Stanney et al. 2002][Iwata and[START_REF] Iwata | Path reproduction tests using a torus treadmill[END_REF], are hardware solutions that enable the user to walk infinitely. But they are often costly, complex and cumbersome. Thus, many software interaction techniques have been developed based on classical 3D input devices such as joysticks or gamepads [START_REF] Bowman | 3D User Interfaces: Theory and Practice[END_REF]]. But they induce very limited physical movements, and sometimes generate a sensation of "flying" in the virtual scene rather than walking [Lécuyer et al. 2006].

The Walking-In-Place (WIP) technique has been introduced by Slater et al. [START_REF] Slater | Taking steps: The influence of a walking technique on presence in virtual reality[END_REF] to enable a real physical walking movement and an efficient navigation technique in 3D virtual environments. The user has to consciously walk in place while motions of his body are tracked and analyzed. The tracked Walking-In-Place motion is used as input for the locomotion simulation inside the VE. First implementations of WIP were all based on the processing of head positions using a neural network. More recent models track the positions of the heels or knees of the user to compute the resulting virtual locomotion [START_REF] Feasel | Llcmwip: Low-latency, continuous-motion walking-in-place[END_REF][START_REF] Wendt | Gud wip: Gait-understanding-driven walking-in-place[END_REF]. However, all existing WIP techniques require the user to stand up, and they focus on immersive VR applications based on sophisticated tracking devices and head-mounted-displays or CAVE setups.

In this paper, we propose to revisit the whole pipeline of the WIP technique. Our intention is to extend it to match a larger set of configurations, by notably applying it to desktop setups. As a result, we propose to improve the WIP technique on three main aspects, corresponding to the three main innovations claimed in our paper:

• Novel user interface capacities. We introduce (1) the possibility for the user to stand or sit (using classical interaction paradigms of desktop VR with mouse), and (2) the possibility to use screens (with limited field of view) and low-cost tracking (webcam) by using the use of head movements as main input of WIP control laws.

• Novel locomotion simulation. We extend the range of possible motions with WIP to: walking, turning, jumping and crawling, using heuristics and control laws based exclusively on users' head movements.

• Novel visual feedback of walking. We introduce the possibility to combine our approach with more realistic visual feedback of walking, i.e. camera motions, that are known to improve the sensation of walking [Lécuyer et al. 2006].

Related Work

Navigating in a virtual environment can be achieved in numerous ways. The most straightforward solution probably consists in walking naturally in the real world, using for instance Head Mounted Display (HMD) together with a tracking system which measures head position for updating viewpoint (or virtual camera position and orientation) in the virtual environment. However, this solution can rarely be used due to the limited physical workspace in the real world, as well as limitations of the tracking devices.

To overcome this problem, numerous "locomotion interfaces" have been designed. the limits of the physical workspace while allowing to navigate infinitely in the virtual world. A classical hardware approach consists in using a treadmill which can be either unidirectional (such as the Sarcos Treadport [START_REF] Stanney | Handbook of Virtual Environments: Design, Implementation, and Applications[END_REF]) or omni-directional (such as the Torus Treadmill [START_REF] Iwata | Path reproduction tests using a torus treadmill[END_REF]). Other locomotion interfaces can be based on feet-platforms [START_REF] Iwata | Gait master: A versatile locomotion interface for uneven virtual terrain[END_REF], walking inside a rotating sphere [START_REF] Fernandes | Cybersphere: the fully immersive spherical projection system[END_REF], or a set of moving tiles [START_REF] Iwata | Circulafloor: A locomotion interface using circulation of movable tiles[END_REF]]. However, the current locomotion interfaces are often too expensive, cumbersome, or complex for being integrated easily in both VR or desktop applications.

Thus, numerous interaction techniques have been developed to provide software-based navigation capabilities in VR. For instance, the World In Miniature (WIM) technique [START_REF] Pausch | Navigation and locomotion in virtual worlds via flight into hand-held miniatures[END_REF] is based on a miniature graphical representation of the VE which can be used to select destinations in the VE. The virtual camera "flies" automatically to the new position selected in the WIM. The Magic Barrier Tape [START_REF] Cirio | The magic barrier tape: a novel metaphor for infinite navigation in virtual worlds with a restricted walking workspace[END_REF]] is another interaction metaphor which enables to navigate infinitely when immersed in a virtual scene, when located in a restricted walking workspace. The physical boundaries of the workspace are displayed in the virtual environment using a barrier tape that must not be crossed by the user but that can be moved in the VE with a joystick.

Besides, classical input devices such as mice, keyboards and joysticks can also be used within interaction metaphors to navigate from one virtual place to another [START_REF] Bowman | 3D User Interfaces: Theory and Practice[END_REF]]. An efficient solution for navigating in 3D VE with a desktop configuration is the interaction paradigm proposed in most First-Person Shooter video games (FPS). It relies on mouse control (for orienting the viewpoint and the direction of motion), and keyboard keys (for forward, backward and lateral motions). However, such interaction techniques, relying exclusively on manipulated input devices, do not provide any proprioceptive or vestibular feedback of walking, and lead to a poor sensation of walking in the VE.

An efficient solution to enhance the sensation of walking in virtual worlds consists in playing with visual feedback and adding artificial camera motions [Lécuyer et al. 2006]. Such camera motions can simulate the visual flow corresponding to natural oscillations of the body and head during the walk. This approach also simulates the ocular reflexes present when walking [START_REF] Hillaire | Using an eye-tracking system to improve camera motions and depth-of-field blur effects in virtual environments[END_REF]]. Camera motions were shown to improve sensation of walking and perception of traveled distances in VE [START_REF] Terziman | Can camera motions improve the perception of traveled distance in virtual environments[END_REF]].

Interestingly, novel commercial systems for optical tracking of body movements have appeared such as the EyeToy for Playstation 2 or, more recently, the Microsoft Kinect . Both systems are focused on interaction with multimedia content and video games but, for the moment, these systems have not been used for direct walking in 3D virtual environments such as when "walking in place".

Walking-In-Place (WIP) is an interaction paradigm designed for both an improved sensation of walking and an efficient navigation in virtual environments [START_REF] Slater | Taking steps: The influence of a walking technique on presence in virtual reality[END_REF]. The WIP technique consists in consciously walking in place in the real world. Previous papers on WIP are presented in Table 1. The first implementation proposed for WIP was based on a HMD. Head tracking was used to predict and detect the steps of the user using a neural network [START_REF] Slater | Taking steps: The influence of a walking technique on presence in virtual reality[END_REF]. When user's steps were detected, the viewpoint automatically moved forward in the VE. The advance direction was simply related to the head's direction. In the next implementations of WIP, the processing of head movements with a neural network was abandoned. The next systems could track the motions of the user's knees or heels. Then, using pattern recognition [START_REF] Templeman | Virtual locomotion: Walking in place through virtual environments[END_REF]], frequency analysis [START_REF] Feasel | Llcmwip: Low-latency, continuous-motion walking-in-place[END_REF] or biomechanically inspired state automaton [START_REF] Wendt | Gud wip: Gait-understanding-driven walking-in-place[END_REF]] the user's steps could be detected and translated into a variable advance speed depending on the steps frequency.

When considering Table 1, it seems that all the WIP techniques described in the literature so far are meant for highly immersive virtual environments and use either HMD or CAVE for visual display [START_REF] Razzaque | Redirected walking in place[END_REF]. These setups use position/orientation tracking of body parts to detect footsteps and compute an advance speed [START_REF] Feasel | Llcmwip: Low-latency, continuous-motion walking-in-place[END_REF]]. The few systems based on head movements are all using neural network as a processing scheme [START_REF] Slater | Taking steps: The influence of a walking technique on presence in virtual reality[END_REF]. Turning in the virtual world is often achieved using the orientation of one of the tracked points (e.g., head, waist, chest) [START_REF] Templeman | Virtual locomotion: Walking in place through virtual environments[END_REF]].

Unfortunately, very few evaluations of WIP have been carried out up to now to compare it with other existing navigation techniques. Moreover, these rare studies have mostly used subjective questionnaires, excepted in [START_REF] Whitton | Comparing VE locomotion interfaces[END_REF]. Considering the data available in the literature, it seems that WIP can increase presence compared to classical joystick-based interaction [START_REF] Razzaque | Redirected walking in place[END_REF][START_REF] Usoh | Walking > walking-in-place > flying, in virtual environments[END_REF]. WIP is also perceived as a more natural technique [START_REF] Yan | New simple virtual walking method -walking on the spot[END_REF]]. However, WIP might also result in an increased cybersickness [START_REF] Usoh | Walking > walking-in-place > flying, in virtual environments[END_REF]. WIP seems also less efficient in term of usability than a real walk [START_REF] Feasel | Llcmwip: Low-latency, continuous-motion walking-in-place[END_REF].

3 Shake-Your-Head: Revisiting WIP for Desktop VR

We propose to revisit the whole pipeline of the Walking-In-Place technique to match a larger set of configurations and apply it to the context of desktop Virtual Reality. Our approach is schematized in Figure 1. The Figure highlights the main differences between our approach and the classical and existing WIP techniques.

With the Shake-Your-Head technique, the user can be standing or sitting (such as in traditional video games or desktop VR configurations). The user interacts with the system by means of head movements. These head movements can be captured using different tracking interfaces, but we insist on the use of low-cost optical tracking with standard webcams. The locomotion simulation proposes not only the computation of a virtual walking motion but also turning, jumping, and crawling possibilities. As a result, the user can perceive the locomotion in the virtual world by means of integrated virtual camera motions on the three axes of motion, to further enhance the sensation of walking.

In the following section, we will successively describe the different parts of our approach, namely: (1) the 3D user interface input/output, (2) the interaction techniques developed for the computation of the virtual locomotion, and (3) the visual feedback relying on camera motions.

Input/Output Interfaces

Our method proposes new features in terms of interfaces in order to extend the set of configurations where WIP can be applied, especially for Desktop VR. Thus, we propose to incorporate new devices for both input and output user interfaces in the VE.

Input: Tracking Based on Head Motions

The input interface of our method is only based on head motions.

In our implementation, the input interface is reduced to a webcam, allowing the use of our method for Desktop configuration without any additional peripheral. However, our method can also be implemented with other classical VR tracking systems.

Use of Head Movements. The main concept of our method is to exploit the head oscillations as a transposition of the one observed during natural walking. While walking, the head of the user oscillates along the lateral, vertical and forward axes [Lécuyer et al. 2006]. The oscillations are strongly correlated to gait events and foot steps. Moreover, these oscillations also occur while walking in place and can be measured.

The head motions are classically retrieved in the existing WIP techniques thanks to the use of regular position trackers [START_REF] Slater | Taking steps: The influence of a walking technique on presence in virtual reality[END_REF]. More generally, any tracking device can be used, as long as its accuracy is within the range of 1cm. Moreover, the acquisition process of the required position does not have any influence, as long as the real time constraint is maintained. In our method, we propose the use of a video camera system to handle the tracking of the user head. Thus, our interaction technique can be deployed on a large scale at low cost for training purpose or video games for example.

Extracted Data. In our method, we propose to use 3 Degrees of Freedom (DOF) that can be easily accessed in the image frame provided by the webcam:

• The lateral position x (and the computed speed Vx);

• The vertical position y (and the computed speed Vy);

• The rotation of the head in the frontal plan r.

These three head motions are illustrated in Figure 2. Implementation. The use of the webcam to track the 3D position of the user head without using markers requires the implementation of real time constraints for the algorithms, i.e. more than 25 frames per seconds. Our implementation is based on the Camshift (Continuously Adaptive Mean Shift) algorithm [START_REF] Bradski | Computer vision face tracking for use in a perceptual user interface[END_REF]] implemented in the OpenCV library. This algorithm is based on color tracking and is well-suited for real-time tracking of features of a given color, such as the face of the user.

While the user is standing in front of the webcam, our algorithm recognizes him as an ellipsoid. The position of the head (x, y) can be deduced from the center of the ellipse and the orientation angle of the head r is given by the angle with the vertical of the ellipse. However, x and y depend on the resolution of the used webcam. Thus, we compute a normalized position (xn, yn) on [-1; 1] on both axes. From the normalized positions of the head (xn, yn), we compute the instantaneous speeds Vx and Vy, and we use Kalman filters on all values to reduce the noise produced by the algorithm.

Output: Immersive and Desktop Visual Displays

Our method can be used with both immersive and regular screens as output interface with the VE. A requirement of desktop VR applications was to propose a technique that is usable with a limited Field of View (FOV). In regular WIP techniques, the provided FOV is always 360 • , except in [START_REF] Razzaque | Redirected walking in place[END_REF] where they used the redirected walking technique to simulate a 360 • FOV in a 4walls CAVE. In this paper, our technique is evaluated with both LCD laptop screens to fulfill the Desktop VR pre-requisite or video projected output. Our method can also be used with the classical output used with WIP, i.e. with HMDs or CAVEs.

Walking Simulation

Walking States

The main goal of our interaction technique is to translate the inputs of the user, i.e. head motions, into virtual motions in the VE. The user should be able to perform various motions while navigating in the VE. We implemented different locomotion states: walking, turning, jumping and crawling. To manage these different states, we added a state automaton to our algorithm (Figure 3). The state transitions are governed by the user head motions and the main inputs are the lateral velocity Vx and the vertical velocity Vy.

Forward State

The forward movements in the VE are governed by the lateral oscillations as main input. Our technique is designed to emphasis the idea of walking with a variating advance speed depending on the user head motions. The advance velocity Va oscillates regularly, accordingly to the lateral head motions. One head oscillation period corresponds to one step. The footstep events are simulated by a null advance speed and correspond to a modification of the lateral velocity sign of the user's head. Thus, when the user's head reaches the maximal amplitude of the oscillations, the oscillating speed is null, as well as the advance speed, simulating a foot step.

For more realistic movements, we introduced two thresholds Tmin and Tmax. Tmin allows to stop the forward movements when the lateral head motions are too small. Tmax allows to avoid unrealistic high speed walks. The advance velocity Va is computed in two steps in order to test these thresholds. The equations of Va are:

Vn 1 = min(abs(Vx),Tmax) Tmax Vn 2 =  0 if Vn 1 < Tmin Vn 1 otherwise Va = Vn 2 * Vmax (1)
Finally, the advance speed V of the camera inside the VE is adapted depending on the current locomotion state and is given by:

V = 8 < : Va if state = walk 0.4 * Va if state = crawl Vmax if state = jump (2)
We chose normalized thresholds with the following values: Tmin = 0.05 and Tmax = 0.5. We also set Vmax to 3.5 meters per second, corresponding to the maximal speed that can be achieved.

Jump and Crawl States

Comparing to existing WIP techniques, we chose to add 2 new states to the navigation possibilities in the VE: Jump and Crawl motions. The jump and crawl states are governed by the vertical oscillations of the user's head. If the vertical velocity exceeds normalized thresholds Tjump in upward direction and T crawl in downward direction, the user can jump and crawl respectively in the VE.

In practice, it means that the user will need to slightly jump or bend forward if he is seated, or jump or crouch down if he is standing.

The user has to stand-up to stop crawling.

When a jump is detected, the vertical position of the camera is set to follow a classical parabolic trajectory defined by:

ycamera = 1 2 * g * t 2 + V * t (3)
with g the gravity acceleration and t the time. The jumping state is left automatically while landing, i.e. when the camera reaches again its normal height H (known as the reference state when the algorithm starts). After preliminary testings, we set Tjump = 0.3 and T crawl = 0.4. While the crawling state is activated, the vertical position of the virtual camera ycamera is lowered by 1 m.

Turn State

In parallel to Forward, Jump and Crawl states, the user has the possibility to turn inside the VE in order to modify his navigation direction. During a turn in a normal walk, the human body leans slightly in direction of the center of the turn to compensate the centrifugal force [START_REF] Courtine | Human walking along a curved path. i. body trajectory, segment orientation and the effect of vision[END_REF]]. This phenomenon is often reproduced by video games players which can tend to lean in the direction of the turn even if it does not have any influence on their in-game trajectory. Thus, we choose to use this property to implement turns in our system as a control law based on the head orientation on the roll axis. To turn in the VE, the user has to lean his head in the left or right side respectively to turn left or right in the VE. The rotation speed Vr of the virtual camera is given by:

Vr = 8 < : Vr max if r > rmax -Vr max if r < -rmax 0 otherwise (4)
where rmax is the minimum angle of head inclination to start the rotation and Vr max is the maximal angular speed of the rotations.

In our experiment, we set rmax = 15 • and Vr max = 45 • /s.

Visual Feedback Based on Camera Motions

To further emphasize the perception of walking in the VE, we extended the visual rendering of the WIP using camera motions driven by the user's head oscillations. There are existing models in the literature that make the virtual camera oscillating along the three axes ([Lécuyer et al. 2006] for example). However, the oscillations are totally independent from the user interactions.

We introduce a new model of camera motions adapted to the user's head motions. The camera oscillations along the different axes must follow the user in real time to maintain the coherency of the system. Thus we have implemented a novel visual feedback with camera motions along the vertical, lateral and advance axes.

Advance Oscillations

The advance speed V of the view point already oscillates. The camera motions are indeed intrinsically linked to the advance velocity of the control law presented in paragraph 3.2.2. As a result, extra camera motion is not necessary along this axis and the advance camera velocity corresponds exactly to V .

Lateral Oscillations

In order to move in the VE, the user has to make his head oscillating from left to right. Thus, as the user moves in front of the screen, his view point of the scene is modified to follow the head oscillations.

The lateral oscillations of the camera are computed as a function of the user's position. If d is the distance of the user to the screen and α and β the opening angles of the webcam, the real world position of the user in front of the screen depends on the normalized coordinates xn and yn. The real world position of the user's head is given by the following coordinates:

 x real = xn * d * tan(α/2) y real = yn * d * tan(β/2) (5)
Finally, the virtual camera is moved along the lateral axis by a distance xcamera equals to: xcamera = Ax * x real . We set the scale factor Ax = 1 to match the user's head displacement and thus generate the illusion that the screen is a window through which the user can observe directly the VE. However, other values can be used to amplify the camera motions for example. The webcam used during the experiment was such as α = 60 • and β = 45 • .

Vertical Oscillations

The vertical oscillations of the camera can not be computed with the same algorithm as for lateral oscillations. In a desktop VR context the user can be seated and not be able to produce high vertical oscillations.

In our method, we propose to generate pseudo-sinusoidal vertical camera oscillations based on the current phase of the virtual gait cycle. Similarly to advance speed control law, the vertical amplitude ycamera of the camera oscillations is given by:

ycamera = Vn 2 * y max camera (6)
where y max camera is the amplitude of the vertical oscillations for velocities greater or equal to the Tmax threshold. For smaller speeds, the amplitude of the oscillations is proportional to this maximum, thus increasing the perception of the variations in advance speeds. Using the same factor between the camera motions and the advance velocity forces the synchronization, resulting in a smooth final visual rendering. In our implementation, we set y max camera = 15cm.

Discussion

To summarize, our approach is composed of (1) an input interface based on the sole user's head movements, (2) a locomotion simulation in the VE composed of various possibilities such as jumping, crawling, turning, and (3) a visual feedback of walking relying on oscillating camera motions. The head motions are tracked along 3 Degrees of Freedom (DOF): lateral, vertical and roll axis (Figure 2). These different physical motions are transposed in virtual movements thanks to a locomotion automaton (Figure 3). We then added oscillating camera motions (Equations 2, 5 and 6) to the visual feedback to enhance the walking sensation. The different control laws were parametrized after preliminary testings. But of course some parameters can be modified in order to amplify/decrease some effects during the locomotion simulation. Besides, other movement possibilities could also be envisaged and added to our automaton such as running state or backward movement.

Evaluation

The evaluation of the proposed technique was performed using a comparison with classical techniques in Desktop VR. We chose keyboard and joystick peripherals for seated and standing positions respectively as they are often used in Desktop VR context. The experiments were conducted using 3D VE displayed on a screen and we investigated the effectiveness of our technique to travel complex paths composed of different gates placed in the VE.

In this paper, we choose to not compare our technique to existing WIP techniques, and instead used more common interfaces that followed our low cost requirement. We conducted the evaluation of the proposed technique in both immersive Standing Up (SU) position and Desktop Sitting Down (SD) position. The keyboard and joystick were chosen respectively as the control conditions in the SD position and the SU position. Both keyboard and joystick peripherals are referred as Control techniques (Ctrl) in the following paragraphs. Our technique is referred with the "WIP" suffix.

Method

Virtual Environment

The evaluation was performed within a 3D virtual environment without any contextual cues. The only landmarks were the gates that the user had to navigate through. A fog effect was added to mask the distant gates, allowing to perceive only the 2 or 3 closest gates. A texture on the ground provided useful visual flow information during the navigation. The scene was normally illuminated, and no shadows were drawn as illustrated in Figure 4.

Population

Twelve participants (10 males and 2 females) aged from 22 to 35 (mean=25.4, standard deviation=3.35) performed the experiment. Two participants were left-handed and none of them had known perception disorders. All participants were used to VEs but were naïve with respect to the proposed technique, as well as to the experimental setup and purpose of the experiment.

Experimental Conditions

The experiment was carried out in a room with controlled luminosity (using two projectors). There were two visual conditions corresponding to the two different configurations (SD and SU). The two configurations are illustrated in Figure 5. For the SU configuration, participants were at a distance of 1.5 m in front of a 1.72 m large and 1.24 m height back-projected screen (physical field of view of 60 • horizontally and 45 • vertically). During the SD configuration, they sat in front of a 17 inches widescreen laptop screen (physical field of view of 30 • horizontally and 18.5 • vertically). In both cases, the resulting image had a resolution of 1600 × 1200 pixels. We used monoscopic rendering, with a frame rate of 60 Hz. The projector used was a DepthQ Stereoscopic.

Experimental Apparatus

In our experiment, the goal was to compare our technique to classical interface devices (keyboard and joystick). During the experiment, the participants had to navigate in the VE through 3 different paths composed of 8 gates each. The 2 first paths, called Normal, were composed of 3 meters by 3 meters wide gates regularly disposed in order to form a slalom. The third path, called Steeple, was composed of 1.5 meters high tunnels and 80 cm high fences (4 of each). For this path, the users had to jump and crawl to pass the gates, while the 2 other paths only required a simple navigation.

The classical interface devices were implemented as follow:

• Keyboard: Forward and turn motions were triggered using the forward, left and right arrow keys. The jumping was triggered by the left Shift key and the crawling by the left Control key.

• Joystick: Forward and turn motions were triggered using the left joystick of a gamepad. Jumping and crawling were triggered with two of the right buttons.

To provide a fair comparison between WIP and Ctrl conditions, the oscillating camera motions were also implemented in the two Ctrl conditions. To do so, we used the best implementation described in [Lécuyer et al. 2006]. Moreover, the Ctrl condition advance speed was set to match the WIP condition average advance speed, which had been measured in a preliminary phase.

The participants had to press the "Enter" key at the beginning of each block of conditions to launch the experiment. At any time during the experiment they had the possibility to make a break by pressing the "Space" key. After each trial, the participants were automatically teleported to the beginning of the next trial. A black screen displayed 2 s notified the beginning of the new trial.

Experimental Plan

The participants were exposed to 4 blocks of 18 trials each: one block for each of the possible combinations of the experimental conditions. The combinations were the following: (1) Sitting Down, using the keyboard (SD-Ctrl); (2) Sitting Down, using the our technique (SD-WIP); (3) Standing Up, using the joystick (SU-Ctrl); (4) Standing Up, using the proposed technique (SU-WIP).

The participants were split in 4 groups equally composed of 3 people each. Two groups started with the SD configuration and the 2 other with the SU configuration. We counter-balanced the conditions between them, meaning that the group starting with Ctrl condition in the SD configuration starts with WIP condition in the SU configuration. For each of the 18 trials of a block, the participants had to navigate through the 3 paths. The 12 first trials were composed only of the normal paths (6 for each, in a random order), and the last 6 trials corresponded to the steeple path.

Collected Data

The WIP techniques are generally evaluated only with subjective questionnaires to evaluate presence or cybersickness. In our evaluation, we added new criteria based on the task performed by the participants. Thus, we measured for each participant the task completion time for each trial and the percentage of success for the different gates. A subjective questionnaire was also proposed.

Results

Task Completion Time

For each participant, the task completion time of each trial was measured for the different experimental conditions. An exploratory analysis was first performed. A Principle Component Analysis revealed the presence of one spurious individual who has been taken out from the analysis. A specific analysis was developed to study the learning effect of the two conditions (Joystick/Keyboard and WIP techniques). A linear model where all conditions are mixed was fitted to explain the relation between the task completion times and the trial number. It revealed that the slope of the linear regression was significantly lower than zero (p-value < 0.000001), reflecting a significant decrease in the task completion time as the number of trials increases. The same analysis, where the first trial was removed, showed that the slope was not significantly different from zero anymore. In the following paragraph, the first trial was removed from the analysis as it corresponds to a learning effect.

A two-way ANOVA was performed on the 2 different conditions (Joystick/Keyboard and WIP techniques) and the 2 different positions (Sit-Down and Stand-Up). A post-hoc analysis using Tukey's procedure was then performed. The ANOVA was achieved separately for the two different types of paths (normal/steeple).

Concerning the normal path, the two-way ANOVA accounting for the conditions and the positions revealed a significant dependency between the position and the task completion time (F (1, 11) = 31.9981, p < 0.0001) and between the condition and the task completion time (F (1, 11) = 6.0449, p = 0.0143). Interaction between condition and position was also considered as a significant factor to discriminate task completion time (F (1, 11) = 27.1921, p < 0.0001). Post-hoc analysis showed that the task completion time in the SD-WIP configuration (M = 55.67s) was significantly lower than in the SD-Ctrl configuration (M = 61.72s), adjusted p-value < 0.0001, in the SU-Ctrl configuration (M = 62.06s), adjusted p-value < 0.0001, and in the SU-WIP configuration (M = 64.23s), adjusted p-value < 0.0001. The other pairs of effects did not give any significant adjusted p-values.

The results concerning the different conditions are represented in Figure 6 for Sit-Down and Stand-Up experiments respectively, for the normal path only. The results are ordered in function of the trials. The first trial was kept to illustrate the learning effect.

For the steeple path, the pre-analysis suggested the presence of a spurious individual and the existence of one learning trial. The twoway ANOVA accounting for the conditions and positions revealed a significant dependency between the position and the task completion time (F (1, 11) = 8.5665, p < 0.005), and the condition and the task completion time (F (1, 11) = 11.8925, p < 0.001. Interaction between condition and position was also considered as a significant factor to discriminate task completion time (F (1, 11) = 8.7647, p < 0.005). Post-hoc analysis showed than the task completion time in the SU-WIP configuration (M = 86.07s) was significantly higher than in the SD-Ctrl configuration (M = 74.78s), adjusted p-value < 0.0001, in the SU-Ctrl configuration (M =

Accuracy

For each participant and for each trial, the percentage of errors for the different paths was measured. The two types of path (Normal and Steeple) are separated. The resulting percentages are for the normal path: 0.61% of error for SD-WIP configuration, 0% of error for SD-Ctrl configuration, 1.39% of error for SU-WIP configuration, 0.09% of error for SU-Ctrl configuration.

The resulting percentages are for the steeple path: 20.49% of error for SD-WIP configuration, 7.81% of error for SD-Ctrl configuration, 27.28% of error for SU-WIP configuration, 13.19% of error for SU-Ctrl configuration. We found a significant effect between Ctrl and WIP techniques for the steeple path.

Subjective Questionnaire

After Sit-Down and Stand-Up configurations, a preference questionnaire was proposed in which participants had to grade from 1 (low appreciation) to 7 (high appreciation) the four different conditions (SD-Ctrl, SD-WIP, SU-Ctrl, SU-WIP) according to 9 subjective criteria: (a) Fun, (b) Easiness of Use, (c) Intuitive, (d) Accuracy, (e) Presence, (f) Walking realism, (g) Fatigue, (h) Cybersickness and (i) Global appreciation. Figure 7 shows the results concerning the grades (Likert-scale) obtained by the two different techniques for each of the subjective criteria, for the two experimental conditions (SD and SU). The grade 7 for Fatigue and Cybersickness respectively means that the technique does not induce any fatigue and does not imply any cybersickness feeling.

Concerning SD configuration, no significant effect was found for the following criteria: Intuitive (p = 0.052) and Cybersickness (p = 0.12). Concerning SU configuration, no significant effect was found for the following criteria: Intuitive (p = 0.3), Walking realism (p = 0.19) and Cybersickness (p = 0.21). We found a significant effect for all other criteria. In particular, our technique was better ranked for Fun, Presence and Global Appreciation, for both configurations.

Discussion

Our results suggest that our technique can allow efficient navigation even compared with standard and well-known input devices such as keyboards and gamepads. The participants could sometimes go even faster with WIP, without any strong loss in precision. The WIP-based interaction seems also fast to learn, after only a couple of trials. The technique is well appreciated and perceived as more immersive and more fun than classical configurations.

The quick learning of our technique could be explained by the fact that interfaces based on webcam are generally intuitive and simple to learn [START_REF] Polaine | The flow principle in interactivity[END_REF]]. After the learning phase, WIP tends to become faster than the keyboard in sitting condition. One explanation could be that with our technique (but also with the joystick in the standing condition) we could observe that participants tended to turn without stopping their advance motion. On the contrary, with the keyboard condition, participants tended to walk and turn sequentially, which might have globally increased the task completion time. Another explanation could be that, in our implementation of WIP, the advance speed is influenced by the speed of lateral movements. The seated position allows the user to make faster oscillations than the standing position, and thus to accelerate the walking motion by making fast lateral oscillations. Interestingly, these faster motions did not impair the precision of users.

The longer task completion time observed for WIP in the steeple paths (involving jumping and crawling motions) could be due to unexpected behaviors which induced incorrect transitions in our locomotion automaton. Indeed, some participants acted as if they were "anticipating" the jumps and bent down prior jumping. We could easily fix this problem in the future by using additional conditions in our automaton based on both speed and position.

Results from the questionnaire are very consistent with previous subjective evaluations of WIP [START_REF] Usoh | Walking > walking-in-place > flying, in virtual environments[END_REF]]. In our study, the WIP is more appreciated, and is perceived as more fun, and improving presence. As expected, more standard techniques (i.e. joystick and keyboard) are found easier to use, more precise, and less tiring (as they induce less physical movements). Interestingly, impression of cybersickness is not increased by WIP. This could be due to our desktop (and thus less immersive) configurations. Last, realism of walking in the VE was significantly improved only in the sitting condition. The perception of walking with WIP is actually quite complex, as participants wrote: "we have the impression to be a video game character", "the motions are exaggerated", or "we really have the sensation of walking, and not running". In the standing condition, some participants found that the physical motions were closer to "skiing" or "skating", as they noticed that they did not lift their feet from the ground but only oscillated their body. For these people we could further stress in the future that our implementation of WIP still works very well when lifting the feet and walking in place, as the oscillations of the head can be captured the same way in both situations (lifting the feet or not).

Taken together, our results suggest that our Shake-Your-Head technique could be used in a wide range of applications, when navigating in a 3D world, in sitting or standing configurations. It seems to be both a low-cost and an efficient paradigm that can match a lot of walking motions. It could thus be used for training in VR with more physical engagement (military infantry, vocational procedures), or more realistic virtual visits such as for project review in architecture or urban planning.

Conclusion

This paper proposed to revisit the whole pipeline of the Walking-In-Place technique to match a larger set of configurations and apply it notably to the context of Desktop Virtual Reality. We have introduced the Shake-Your-Head technique using as sole input the head movements of the user. It can be used in a desktop configuration with the possibility for the user to sit down and to navigate in the VE through small screens and standard input devices such as a basic webcam for tracking. We implemented various motions such as turning, jumping and crawling in the locomotion simulation. We also introduced the use of additional visual feedback based on camera motions to enhance the walking sensation.

We conducted an experiment in order to evaluate our technique compared to standard techniques such as keyboard and joystick. In this experiment, participants had to walk through a series a gates forming a slalom path. The evaluation was performed both in an immersive and desktop configurations. We notably found that WIP technique only requires a small learning time to allow faster navigation in seated position compared to the keyboard. Moreover, our technique was more appreciated and considered as more fun and inducing more presence than the other classical techniques.

Future work will focus on improving the underlying locomotion simulation. We could add new features such as running or backward movements in the VE. We could also improve the control laws with a more biomechanically-inspired description. Second, we also plan to conduct more evaluations, especially for further studying the perceptive aspects and measuring the influence of WIP on estimating spatial properties of VE.

Figure 1 :

 1 Figure 1: Overview of our novel approach for Walking-In-Place (one star stresses improvement of existing component, two stars stress additional components).

Figure 2 :

 2 Figure 2: Extracted head motions: (a) lateral motion, (b) vertical motion, (c) roll motion.

Figure 3 :

 3 Figure 3: Walking state automaton: the main inputs are the velocities Vx and Vy. Vx allows the transition from the Stop state to the Forward motion; Vy allows the transition to the Jump or Crawl states. The Turn motion is not represented as it can be activated in any state, depending on the head orientation.

Figure 4 :

 4 Figure 4: Virtual environment with different gates to represent the path that the user has to navigate through.

Figure 5 :

 5 Figure 5: Description of the experimental setup for (a) the Stand-Up (SU) configuration, and (b) the Sit-Down (SD) configuration.

Figure 6 :

 6 Figure 6: Task completion time for the two different techniques on normal paths for (a) SD experiments and (b) SU experiments. The blue and red light colors correspond to the Ctrl and WIP conditions respectively. Each boxplot is delimited by the quartile (25% quantile and 75% quantile) of the distribution of the condition over the individuals. The median is also represented for each condition.

Figure 7 :

 7 Figure 7: Results for subjective rating for the two different techniques for (a) SD experiments and (b) SU experiments. The blue and red light colors correspond to the Ctrl and WIP conditions respectively. The subjective criteria are (a) Fun, (b) Easiness of Use, (c) Intuitive, (d) Accuracy, (e) Presence, (f) Walking realism, (g) Fatigue, (h) Cybersickness and (i) Global appreciation. Each boxplot is delimited by the quartile (25% quantile and 75% quantile) of the distribution of the condition over the individuals. The median is also represented for each condition.

Table 1 :

 1 Overview of existing Walking-In-Place techniques.

			Input/Output Interfaces		Interaction Technique
	References	Output: Display Input: Tracking	Control Law	Available Motions
	[Slater et al. 1995]	HMD	Head	Neural network	FW, Turn
	[Usoh et al. 1999]	HMD	Head	Neural network	FW, Turn
	[Templeman et al. 1999]	HMD	Knees, waist, hands, head	Knee pattern	FW, BW, Diagonal, Sidestep, Turn
	[Razzaque et al. 2002]	CAVE	Head	Neural network	FW, Turn
	[Yan et al. 2004]	CAVE	Legs, waist, head	Leg speed relation	FW, Turn
	[Feasel et al. 2008]	HMD	Shins	Filtered heels speed	FW (various speeds), Turn
	[Wendt et al. 2010]	HMD	Shins	Biomechanical state machine FW (various speeds), Turn

The locomotion interfaces keep the user within


Isabelle Cantat

Benjamin Dollet

Liquid films with high surface modulus moving in tubes: dynamic wetting film and jumpy motion

Keywords: numbers: 47.15.gm, 47.55.dk, 82.70.Rr

We investigate the motion through a wet tube of transverse soap films, or lamellae, of high surface dilatationnal modulus. Combining local thickness and velocity measurements in the wetting film, we reveal a zone of several centimeters in length, the dynamic wetting film, which is significantly influenced by a moving lamella. The dependence of this influence length on lamella velocity and wetting film thickness provides a discrimination among several possible surfactant minimal models. A spectacular jumpy mode of unsteady motion of a lamella is also evidenced.

The motion of a liquid meniscus sliding over a wet wall is ubiquitous in many industrial and biological contexts, among which dip coating [START_REF] De Gennes | Gouttes, bulles, perles et ondes[END_REF], bubble or droplet motion in pores (e.g. in enhanced oil recovery) and microfluidic channels [START_REF] Bruus | Theoretical Microfluidics[END_REF], foam friction on solid boundaries [START_REF] Denkov | [END_REF], as well as lung diseases [4]. The problem has first been studied by Landau and Levich [5] and Derjaguin [6] (LLD) for a solid plate pulled out of a liquid bath at small velocity. For pure liquids, the dynamics is controlled by the dynamic meniscus that forms between the static meniscus and the wetting film withdrawn by the plate. The LLD model leads to a wetting film thickness h LLD = 0.945ℓ c Ca 2/3 , with ℓ c the capillary length and Ca = ηU/γ the capillary number (γ: surface tension, η: liquid viscosity, and U : plate velocity); the extension of the dynamic meniscus along the plate scales like L LLD ∼ ℓ c Ca 1/3 . These predictions are based on the lubrication approximation h LLD /L LLD ≪ 1, and on a free shear boundary condition at the air/liquid interface.

If the liquid phase is a solution of surfactants, the interfacial stress depends on the surface concentration of surfactants Γ and on the rheology of the surfactant layer. This stress reacts against the local area variation of the interface, with a dependence quantified by a surface dilatational modulus E. The friction against a wall of foams made of solutions with high E is well predicted by the limiting case of a locally incompressible interface [START_REF] Denkov | [END_REF]: the whole interface then moves with the foam, resulting into a strong shearing of the wetting film between the foam and the wall. For single bubbles and foams, the total interface area of each bubble is constant during the motion, which is compatible with the local incompressibility of the interface.

In other geometries, like dip coating or motion of a lamella (a soap film across a tube), the total area of each connected part of the interface varies, and the location and extension of the zone where area variations occur becomes a central question. A classical assumption is that the area variations are localized in the static meniscus, and that, in contrast with the model by Denkov et al. [START_REF] Denkov | [END_REF], the interface of the wetting film and of the dynamic meniscus move with the plate. The hydrodynamical problem is then governed by the same equations as for pure liquids, after a well adapted rescaling [7]. Several models go beyond these limiting cases, by including surfactant adsorption/desorption [8], diffusion in the bulk [9] or an intrinsic surface viscosity [10]. Recently, the full problem was numerically solved without the lubrication approximation, which allowed to include the whole meniscus into the simulation, with surface diffusion [11] or surface diffusion and adsorption/desorption [12]. However, no clear experimental evidence allows to discriminate between these different models.

Using a surfactant solution with high surface modulus and low bulk viscosity, we bring the first direct experimental evidence that a lamella moving in a wet tube can push the wetting film over centimetric distances, i. e. more than two orders of magnitude larger than L LLD . We thus introduce the concept of dynamic wetting film. We show that its lateral extension, that we call the influence length, depends on the initial film thickness and on the lamella velocity. Moreover, we show that the various processes that may govern the influence length lead to very different scalings, discriminated by our experimental data. Finally, we demonstrate a surprising jumpy behavior at high velocity, where the meniscus intermittently slips over its wetting film.

Single lamellae are created and pushed at prescribed velocity U in a wetted vertical tube of inner diameter 2a = 8.8 mm following the method of [13]. In order to maximize E, we use the well characterized mixture of sodium lauryl dioxyethylene sulfate, cocoamidopropyl betaine, and myristic acid (MAc) in ultrapure water (solution S 1 ), or the same with 40% wt glycerol (solution S 2 ), following the protocol of [14]. The wetting film thickness h is measured by white light interferometry using the commercial spectrometer USB4000 (Ocean Optics). Neglecting multiple reflections within the film, the collected light intensity obeys I(λ) = I 0 [α + β cos(4πnh/λ)] with I 0 a reference intensity, n = 1.33 the optical index of the solution and λ the wavelength. The film thickness h fib (t) is determined by computing the dominant Fourier component of I(1/λ). Assuming a steady film profile in the lamella frame, we set h(x) = h fib (tx/U ), with t 0 the time at which the lamella is in front of the fiber. The direction x is oriented downstream, with x = 0 at the lamella position (see Fig. 1, right). A camera is synchronized with the spectrometer and records the film shape and velocity (Fig. 1, left). Typical wetting film profiles are given on Fig. 2. The signal modulation is destroyed for too high thickness gradients, hence the meniscus profile is not accessible. The wetting film thickness h 0 , far ahead the lamella, depends on the waiting time since the previous lamella. Its evolution by drainage is slow enough for h 0 to remains uniform and constant on the experimental space and time scales, in the range 1 to 20 µm. The thickness h 1 of the film deposited by the lamella does not need to be equal to h 0 . If h 0 = h 1 , the meniscus volume changes, but slowly enough for the steady state assumption to remain valid.

Most remarkably, the wetting film begins to swell several centimeters in front of the lamella, at a distance L 0 . Some profile shapes are roughly exponential, but the thickest ones display a sharp transition between the flat wetting film and the swollen part of the film close to the meniscus, of typical thickness h ≈ 2h 0 . The rear influence length L 1 is significantly smaller, but still centimetric. We compute L 0 and L 1 a systematic fitting procedure leading to the characteristic distances marked on Fig. 2. L 0 is plotted as a function of U and h 0 on Fig. 3. The data are somewhat scattered, but exhibit a clear tendency: L 0 increases with h 0 , and decreases with both η and U ; the best power law fit yields:

L 0 ∝ h 1.0±0.1 0 U -0.5±0.1 η -1.2±0.1 . (1)
A similar law was obtained for L 1 (data not shown).

In some experiments, the presence of tiny bubbles within the wetting film enabled the simultaneous measurement of the film thickness and of the surface velocity. These bubbles are convected at a velocity ūb = αū s , with ūs the surface velocity (the bar indicating velocities in the laboratory frame) and α 1 an unknown friction parameter. Comparison with the film thickness profile proves that, as the film begins to swell, the velocity rapidly increases from almost zero to a value of the order of U (Fig. 2, inset). The velocity gradient in the dynamic wetting film is thus close to U/h, and the friction force per unit length (in the z direction) can be estimated as

f v ≈ ηU (L 0 /h 0 + L 1 /h 1 )
. This estimate, in the range 10 to 30 mN/m for all our experiments, is in good qualitative agreement with the value extracted from the film images (Fig. 1) f v = 2γ 0 a/R [13], with γ 0 = 23.8 mN/m the equilibrium surface tension [14] and R the radius of curvature of the lamella (data not shown). The friction in the dynamic meniscus alone is given by the Bretherton law f v,B = 4.70γ 0 Ca 2/3 [17] and is at most 1.6 mN/m in our range of velocities. It is thus negligible compared to the dynamic wetting film contribution for this solution.

We also tested a SDS solution, with η = 1.2 mPa•s and E < 1 mN/m and a solution of 12-hydroxy stearic acid with ethanolamine as a counterion [18], with E = 38 mN/m and a rheothinning behavior with a high viscosity η varying between 20 and 10 -2 Pa.s for shear rates between 10 -2 and 300 s -1 In both cases, no measurable influence length has been observed. This gives a strong hint that a large L 0 is associated to solutions of high E and low η. The influence length results from a competition between the resistance of the surface against compression, and the viscous resistance of the bulk against shear. The former may arise from various microscopic mechanisms, governed by the surface viscosity or elasticity, coupled with the surfactant desorption rate or diffusion. We now investigate them in their simplest form, at the expense of a quantitative modeling; however, this scaling approach appears to be sufficient to identify only one scenario able to capture the main dependences experimentally observed, in the investigated parameter range.

We model the region x > 0, ahead of the lamella, in the frame of the lamella. We consider a steady regime. Since h ≪ a, the tube curvature is negligible and we assume an invariance in the z direction. In the dynamic wetting film but at a distance larger than L LLD from the lamella, Marangoni effects dominate over capillary effects and we can neglect the Laplace pressure. The velocity u(x, y) is thus oriented along x and varies linearly between -U at y = 0 and u s (= ūs -U ) at y = h. Mass conservation thus writes 0 = ∂ x [(u s -U )h/2], hence:

u s = U 1 - 2h 0 h , (2)
in agreeement with experimental data (see Fig. 2,inset).

In the presence of the surface viscosity µ * , the tangential stress continuity imposes, at y = h(x) [START_REF] Edwards | Interfacial Transport Processes and Rheology[END_REF],

η∂ y u = ∂ x γ + µ * ∂ xx u s . (3)
Ref. [14] shows that without MAc, the surface modulus is two orders of magnitude lower. Therefore, it is reasonable to assume that ∂ x γ is only related to ∂ x Γ MAc , hence to use a one-component model. Assuming for simplicity a linear relation for the surface tension γ = γ 0 -E(Γ -Γ 0 )/Γ 0 with Γ 0 the equilibrium surface concentration in MAc and Γ = Γ MAc , we get the velocity:

u(x, y) = 1 η µ * ∂ xx u s - E Γ 0 ∂ x Γ y -U. ( 4
)
If the interfacial dynamics was dictated by surface viscosity rather than surface elasticity, Eq. ( 4) would become, for y = h and E = 0:

1 + u s /U = (µ * h/ηU )∂ xx u s .
From experiments, the variation of u s is of the order of U (Fig. 2, inset), leading to the scaling:

L 0,v = µ * h 0 η 1/2 , (5)
which does not capture the observed dependence of L 0 on the lamella velocity. We now neglect the interfacial viscosity to focus on the other contributions from the surfactants. The surfactant mass balance at the interface writes:

∂ x (u s Γ) = j, (6)
with j the surfactant flux from the bulk to the interface. The dynamic wetting film is compressed by the moving lamella, inducing a typical surface concentration variation ∆Γ > 0. Moreover, Eq. ( 4) leads to:

u s ∼ -U -Eh ηΓ0 ∂ x Γ, from which we deduce u s ∂ x Γ ∼ U ∆Γ L0 and Γ∂ x u s ∼ -∆Γ L0 Eh0 ηL0
. Experimentally, we get Eh0 ηUL0 ≈ 3 with h 0 ≈ 10 -5 m, U ≈ 1 cm/s and L 0 ≈ 3 cm, showing, as expected, that the convective term ∂ x (u s Γ) is dominated by the velocity variation, which tends to accumulate surfactant in front of the lamella. The concentration gradient only reduces this effect. If we neglect this last term, we get the following scaling law for the convective term in (6), as already obtained in [8]:

∂ x (Γu s ) ∼ - Eh 0 ∆Γ ηL 2 0 < 0. (7)
This term is balanced by the exchange term with the bulk j. It equals the diffusive flux in the subphase, and obeys a kinetic sorption law [START_REF] Edwards | Interfacial Transport Processes and Rheology[END_REF] that we assume linear. Then, j = -D∂ y c = k(c -Γ/h Γ ) at y = h, with c the bulk concentration in MAc, c 0 its equilibrium value, h Γ = Γ 0 /c 0 , D the diffusion coefficient, and k the sorption velocity, both assumed to be the ones of the micelles in which MAc is solubilized [14]. If k is small, the exchange is limited by the desorption process, and c ≃ c 0 . Then j ∼ -k∆Γ/h Γ , hence from ( 6) and ( 7):

L 0,a ∼ Eh 0 h Γ ηk 1/2 . ( 8
)
This scaling is exactly the same as in the viscous case (5), with an effective surface viscosity µ * eff = Eh Γ /k. If the desorption is fast, the surface is in equilibrium with the subphase and, close to the meniscus, the concentration increase in the subphase scales as ∆c(h) = ∆Γ/h Γ . Then, the flux directly depends on the bulk diffusive field: j = -D∂ y c. If the gradient is established over the diffusive distance h d ∼ DL 0 /U , this leads to the scaling behavior:

L 0,d ∼ Eh Γ η 2/3 h 2/3 0 (DU ) 1/3 . ( 9
)
This is only valid if h d < h 0 . Injecting (9) in the expression of h d , we get the condition K = [EDh Γ /(ηU 2 h 2 0 )] 1/3 < 1. With the experimental parameter values (see caption of Fig. 3), we get K ≈ 2. If h d ≫ h 0 , the gradient ∂ y c becomes small and j is estimated from surfactant conservation: j = ∂ x ( h 0 ucdy) ≃ ∂ x (c h 0 udy). From mass conservation, the flow rate h 0 udy is a constant, equal to -U h 0 Hence, j ∼ -U h 0 ∆c/L 0 , and L 0 in this convective regime is thus governed by:

L 0,c ∼ Eh Γ ηU , (10)
which does not capture the observed dependence of L 0 on h 0 . Overall, the only regime compatible with our experimental results is the diffusive one. Even if the exponents in ( 1) and ( 9) slightly differ, the rescaling by ( 9) shown in Fig. 3 (inset) provide a reasonable agreement with the experimental data, without adjustable parameter. Moreover, we do not explain why L 1 is significantly shorter than L 0 (Fig. 2); this suggests that the interface resists more compression than extension. Finally, the most deformed lamellae deviate from steady motion by a striking scenario: they undergo periodic "jumps" between phases of constant velocity (Fig. 4). These jumps are quick (less than 10 ms) and macroscopic (of order 1 mm), which make them easily observable by naked eye. The curvature is partially released during jumps, and builds up again during the phases of constant velocity. Integrating Eq. ( 3) (with µ * = 0) along the front part of the dynamic wetting film yields γ -= γ 0 -∞ 0 η∂ y udx ∼ γ 0 -ηU L 0 /h 0 , with γ -the value of the surface tension close to the meniscus. The largest values obtained for U L 0 /h 0 are close to γ 0 , which shows that surface tension significantly decreases towards the meniscus. In this case, the linear relation between γ and Γ does not hold anymore, which may explain the discrepancy between the experimental and theoretical exponents. It is likely that the interface becomes unstable below a certain value of γ -and collapses as in Langmuir monolayers [START_REF] Lee | [END_REF], and that the jumps are macroscopic manifestations of such an instability. This unsteady behavior is in marked contrast with that of SDS films, which never showed such jumps [13].

As a conclusion, our predictions of the length over which an air/liquid interface of a wetting film is entrained by a lamella may provide a quantitative criterion for the transition between tangentially immobile and mobile interfaces in foam/wall friction [START_REF] Denkov | [END_REF]. With ℓ a typical bubble size, these two limits correspond respectively to L 0 ≫ ℓ and L 0 ≪ ℓ, and the transition criterion L 0 = ℓ may be expressed from our predictions, in terms of the material parameters of the surfactants. Experimental tests of this hypothesis are under way.

We thank A. Saint-Jalmes, N. D. Denkov and S. Tcholakova for enlightening discussions about solution properties.

FIG. 1 :

 1 FIG. 1: (left) Lamella passing in front of the fiber. (right) Scheme of the meniscus and notations used in the text.

FIG. 2 :FIG. 3 :

 23 FIG.2: Wetting film profiles obtained with different initial thicknesses and U = 0.6 cm/s (solution S1). The values obtained for Li and hi (i = 0, 1) after profile analysis are indicated resp. by thick segments and dashed lines. The central box corresponds to the meniscus and x = 0 is the lamella position. Inset: Film profile obtained with U = 1.7 cm/s, S1 (thick line), ūs measured by tracer tracking for the same experiment, using α = 0.45 (dotted line) and ūs deduced from the film profile using Eq. 2 (thin line).

FIG. 4 :

 4 FIG.4: Time evolution of the position of the meniscus (plain line) and of the leading point of the lamella (dashed line), in the laboratory frame and in the jumpy regime. Two snapshots show the shape of the lamella just before, and just after, a jump.


R Blach

M. D S Coquillart

A 'cruz

G Steed

Welch

Léo Terziman

Maud Marchal

Franck Multon

Bruno Arnaldi

Anatole Lécuyer

Joint Virtual Reality Conference of EuroVR -EGVE (2011)

Keywords: (according to ACM CCS): Computer Graphics [I, 3, 7]: Three-Dimensional Graphics and Realism-Virtual Reality; Computer Graphics [I, 3, 6]: Methodology and Techniques-Interaction Techniques

In this paper we analyze and compare the trajectories made in a Virtual Environment with two different navigation techniques. The first is a standard joystick technique and the second is the Walking-In-Place (WIP) technique. We propose a spatial and temporal analysis of the trajectories produced with both techniques during a virtual slalom task. We found that trajectories and users' behaviors are very different accross the two conditions. Our results notably show that with the WIP technique the users turned more often and navigated more sequentially, i.e. waited to cross obstacles before changing their direction. However, the users were also able to modulate their speed more precisely with the WIP. These results could be used to optimize the design and future implementations of WIP techniques. Our analysis could also become the basis of a future framework to compare other navigation techniques.

Introduction

In Virtual Reality (VR) applications, navigation is one of the fundamental 3D interaction tasks [START_REF] Bowman | 3D User Interfaces: Theory and Practice[END_REF]. The ability to walk is one of the most basic and essential need to complete other more complex tasks within Virtual Environments (VE) (Figure 1). However, most of the time, the working space is limited and real walking cannot be used. Various techniques have been developed so far, based on specific devices such as treadmills or on software metaphors.

The Walking-In-Place (WIP) technique [SUS95] has been introduced to enable a real physical walking movement during virtual navigations. With WIP, the user can move infinitely inside the VE by consciously walking in place in the real world. The motions of the user's body are tracked and analyzed to be converted into inputs for the † firstname.lastname@inria.fr . However, the navigation techniques are usually used in complex navigations, including turns and changes of speed. To the authors' best knowledge, the WIP has never been evaluated on such kind of trajectories so far. In this paper, we analyse and compare the trajectories produced by the WIP with ones produced by joystick on slalom paths.

Experimental Data

The trajectories of 2 navigation techniques are compared: the WIP technique and a standard navigation using a joystick. We chose to implement the WIP technique proposed in [TME * 10] using a webcam-based tracking (25 Hz capture), while the joystick was based on a classical analogical implementation. For the WIP technique, the advance speed was computed based on the lateral amplitude and frequency of the user's head oscillations. Both techniques were calibrated to provide the same average speed during the navigations. Because of the limited Degrees of Freedom of the tracking system, the orientation of the head on the roll axis was used to turn. For both techniques, the turns were based on a rate control law of 45 • /s. The WIP implementation, as well as the experimental apparatus used are described with more details in [TME * 10]. During the experiment, the participants had to navigate in the VE through 2 different paths composed of eight 3m x 3m gates regularly placed composing a slalom. The instruction received by the participants was to cross the gates in the correct order.

Population: Twelve participants (10 males and 2 females) aged from 22 to 35 (Mean(M) = 25.4, Standard Deviation(SD) = 3.35) performed the experiment. All participants were used to VE but were naïve with respect to the WIP technique and the experimental setup. Apparatus: The evaluation was performed within a 3D virtual environment without any contextual cues, except for gates that the user had to navigate through. A fog effect was added to mask the distant gates, allowing to perceive only the 2 or 3 closest gates. A texture on the ground provided visual flow information during the navigation. Participants were at a distance of 1.5 m in front of a 1.72 m large and 1.24 m height back-projected screen (with a physical field of view of 60 • horizontally and 45 • vertically). The resulting image had a resolution of 1600 × 1200 pixels. Experimental plan: The participants were exposed to 2 blocks of 12 trials each: one block for each of the experi-mental conditions. Half of the participants started with the joystick condition, while the other half started with the WIP. Collected Data: The trajectories of the participants in the VE were recorded at a frequency of 60 Hz for every trial. The orientation of the virtual camera was also recorded.

Trajectories Analysis

The proposed evaluation is based on both spatial and temporal analysis. Moreover, the spatial analysis focuses on both macroscopic and microscopic aspects of trajectories.

Spatial Analysis

The produced trajectories are time parametric functions. However, some relevant information can be extracted from the shape of the trajectories, independently of time.

Distance to Shortest Trajectory:

To characterize the differences between the two conditions, we computed for each participant the mean difference between the produced trajectories and the shortest trajectory for each condition. The shortest trajectory was defined as the trajectory composed of straight lines between the inner post of each gate. We conducted a one-way repeated measure ANOVA on the technique used (joystick and WIP). The ANOVA revealed a significant dependency between the distance to the shortest trajectory and the interaction technique used (F(1, 11) = 16.94, p = 0.002). The area between the trajectory and the shortest trajectory is significantly smaller with the joystick technique (M = 86.55 m 2 ) compared to the WIP technique (M = 110.93 m 2 ).

Mean Trajectory: To further stress the differences between the two techniques, we computed the mean trajectory for all participants for each set of conditions for the two paths. We re-sampled the trajectories with samples every 1m in the forward direction. The new samples were computed as the mean of all the values in the sample interval. A sample of the mean trajectories for the joystick and WIP conditions on the first path is displayed in Figure 2. The standard deviations of the mean trajectories represented in Figure 2 stress higher variability of the trajectories produced with the WIP technique. Moreover, the behaviour during the crossing of the gates seems to be different.

For each gate of the 2 mean trajectories, we computed the signed distance of the inflexion point to the closest gate, as well as the distance to the center of the gate when the users crossed the gates. For both criteria, we conducted a one-way repeated measure ANOVA on the technique used (joystick and WIP). We found a significant effect of the technique on the position of the inflexion points (F(1, 13) = 17.33, p = 0.001). With the WIP technique the inflexion point is located after the gate (M = 0.14 m), while it is located before with the joystick technique (M = -0.42 m). We also found a significant effect of the technique on the distance to the center of the gate during the crossings (F(1, 13) = 34.75, p < 0.001). Thus, the user crossed the gates closer to their centers with the WIP technique (M = 0.38 m) compared to the joystick (M = 0.63 m). In other words, with the WIP technique, the participants turned after crossing the gates, while they turned before with the joystick.

Curvature of the Trajectory: In order to further stress the differences in the behavior during the turns, we focused our analysis on the curvature of the trajectories. We computed the curvature at a microscopic level of trajectories parametrically defined as plane curves over time. This elicits a high difference in magnitude of curvature peaks between WIP and joystick. The Figure 3 illustrates this difference for two trials of the 6 th participant. The signal made of the curvatures of the trajectory over time is composed of impulses. In order to count and detect impulses, we used a first order continuous-time low-pass filter with a cut-off frequency of 10 Hz to filter the computed curvatures, and set a detection threshold for impulses at 0.02 m -1 .

To characterize the properties of the curvature for the different conditions, we computed 3 different criteria based on the detected impulses: (1) the number of impulses per trial, (2) the length of the impulses (in seconds) and (3) the area of each impulse. For each criterion, we computed the mean for each participant and we conducted a one-way repeated measure ANOVA on the technique used (joystick and WIP). The analysis revealed a significant dependency to the technique used for all 3 criteria. The results are summarized in Table 1. The participants significantly turned more often with the WIP technique (M = 1108.9, SD = 284.6) than with the joystick (M = 608.9, SD = 77.2). However, they turned significantly for a shortest amount of time with the WIP (M = 0.33 s, SD = 0.08 and M = 0.45 s, SD = 0.07 respectively). Finally, the area of impulses revealed that the participants significantly turned more with the WIP technique (M = 0.19 s.m -1 , SD = 0.14) than with the joystick (M = 0.08 s.m -1 , SD = 0.02).

Criterion F(1,

Temporal Analysis

Speed: We computed the advance speed of the participants depending on the experimental conditions. We filtered and re-sampled the data using the same method as with the computation of the mean trajectories. To characterize the speed behaviour during the trajectories, we computed the average speed for all participants at each gate. We conducted a one-way repeated measure ANOVA on the technique used. We found a significant effect of the technique used on the speed at each gate (F(1, 13) = 195.46, p < 0.001). The users were faster with the WIP technique (M = 2.48 m.s -1 , SD = 0.1) compared to the joystick technique (M = 2.10 m.s -1 , SD = 0.007). These results suggest that participants tended to cross the gates at full speed with the WIP technique.

Smoothness:

The maximum smoothness model [PHA * 07] was used as another criterion. According to this model, natural motions tend to be as smooth as possible to c The Eurographics Association 2011. minimize the energy involved. For each participant, we computed the mean value of the integrated absolute value of the jerk over the time for each trajectory. Greater jerk corresponds to a smaller smoothness of the trajectory and thus can be considered as less natural. A one-way repeated measure ANOVA on the technique used (joystick and WIP) revealed a significant dependency between the cumulated jerk and the technique used (F(1, 11) = 59.56, p < 0.001). The trajectories are smoother with the joystick technique (M = 10.7 m.s -2 , SD = 1.6) compared to the WIP technique (M = 197.5 m.s -2 , SD = 84.0). Thus, the trajectories produced by the joystick seem to be closer to realistic natural walking trajectories compared to WIP, according to the smoothness model [PHA * 07].

Discussion

Our results show strong differencies between trajectories produced by the WIP and the joystick, according to different criteria. The characteristics of the curvatures highlight more frequent and tighter changes in direction with the WIP technique. It seems that the users were not able to predict their future trajectory accurately with the WIP technique, resulting in a continuous adaptation of the advance direction. The position of the inflexion points and the distance to the center of the gates at each gate are consistent with this hypothesis. The implementation of the turns with the WIP might explain those results. Indeed, if the user's head oscillations were too important, turns could be triggered inappropriately. Moreover, our results suggest that the trajectories produced with the WIP technique are less "efficient" in terms of traveled distances as well as naturalness compared to those made using the joystick. This difference could be explained by the difference in training of the participants. Indeed, all of them were proficient with the joystick but were using WIP for the first time. The control law for turns can be another reason, as the participants were apparently less able to choose their direction accurately with the WIP technique.

Finally, the curvature of the trajectories is clearly more important with the WIP technique. The participants tended to follow tighter turns. However, the underlying implementation of the turns for both joystick and WIP conditions were strictly identical. Thus, the participants decreased their speed during the turns with the WIP. The users seem more able to modulate their speed precisely with the WIP, which provided more varied navigation patterns. These results could provide guidelines for future implementations of turns when designing a WIP technique.

Conclusion

In this paper, we compared a WIP technique to the joystick interface. When observing slalom trajectories produced with both techniques, we found that the navigation strategy of the user was different with the WIP technique compared to the joystick. The participants had more difficulty to anticipate their trajectory with the WIP. We also found that the speed during the turns decreased and the user modulated their speed more precisely with the WIP technique. The trajectories produced by the WIP had more jerk than those produced by the joystick and thus were less likely to feel natural for the users. However, the WIP provided a better control of advance speed while the joystick was more precise for controlling direction.

Taken together, our results provide insights on the behavior of users when navigating using WIP vs. joystick. They could inspire guidelines for future design and implementations of WIP techniques, notably concerning the handling of turns. Last, a future framework to compare 3D navigation techniques could be built based on the proposed analysis.

Figure 1 :

 1 Figure 1: Slalom navigation in a Virtual Environment.

Figure 2 :

 2 Figure 2: Samples of the mean trajectories of WIP and joystick. The error boxes represent the standard deviations. Gates are represented in green, the distance to the shortest trajectory in gray and the inflexion points in red.

Figure 3 :

 3 Figure 3: Curvatures of trajectories for two trials of the 6 th participant.

Table 1 :

 1 Results of the ANOVA on the curvature criteria.

		11)	p
	Number of impulses	30.27	< 0.001
	Length of impulses	13.43	0.004
	Area of impulses	6.77	0.025

  10] TERZIMAN L., MARCHAL M., EMILY M., MULTON F., ARNALDI B., LÉCUYER A.: Shake-your-head: Revisiting walking-in-place for desktop virtual reality. In Proceedings of the ACM Symposium on Virtual Reality Software and Technology (2010), pp. 27-34. 1, 2 [UAW * 99] USOH M., ARTHUR K., WHITTON M. C., BASTOS R., STEED A., SLATER M., BROOKS JR. F. P.: Walking > walking-in-place > flying, in virtual environments. In Proceedings of SIGGRAPH (1999), pp. 359-364. 2 [WCF * 05] WHITTON M. C., COHN J., FEASEL J., ZIMMONS P., RAZZAQUE S., POULTON S., MCLEOD B., JR F. B.: Comparing VE locomotion interfaces. In Proceedings of IEEE Virtual Reality (2005), pp. 123-130. 2 [WWB10] WENDT J., WHITTON M. C., BROOKS F.: Gud wip: Gait-understanding-driven walking-in-place. In Proceedings of IEEE Virtual Reality (2010), pp. 51 -58. 1 [ZLB * 05] ZANBAKA C., LOK B., BABU S., ULINSKI A., HODGES L.: Comparison of path visualizations and cognitive measures relative to travel technique in a virtual environment. IEEE Trans. on Visualization and Computer Graphics 11 (2005), 694-705. 2

	123-
	130. 2
	[SUS95] SLATER M., USOH M., STEED A.: Taking steps: The
	influence of a walking technique on presence in virtual reality.
	ACM Trans. on Computer-Human Interaction 2, 3 (1995), 201-
	219. 1
	[TME

[PHA * 07] PHAM Q., HICHEUR H., ARECHAVALETA G., LAU-MOND J., BERTHOZ A.: The formation of trajectories during goal-oriented locomotion in humans. II. a maximum smoothness model. Eu. J. of Neuroscience 26, 8 (2007), 2391-2403. 3, 4 [RSS * 02] RAZZAQUE S., SWAPP D., SLATER M., WHITTON M. C., STEED A.: Redirected walking in place. In Proceedings of the EGVE workshop on Virtual Environments (2002), pp. * c The Eurographics Association 2011.

c The Eurographics Association 2011.

Acknowledgment:

The authors would like to thank Dr Géry Casiez for his help on the data analysis.


David Hagenmüller

Cristiano Ciuti

Cavity QED of the graphene cyclotron transition

Keywords: numbers: 42.50.Pq, 78.67.Wj, 71.70.Di

published or not. The documents may come

Cavity Quantum Electrodynamics (QED) in the ultrastrong coupling regime is a fascinating topic that is attracting considerable interest in condensed matter physics, particularly in semiconductor microcavities [START_REF] Haroche | Exploring the quantum: atoms, cavities, photons[END_REF][START_REF] Ciuti | [END_REF][3][4][5] and superconducting circuit QED systems [6][7][8][9]. The ultrastrong coupling regime is achieved when the vacuum Rabi frequency (quantifying the interaction between one cavity photon and one elementary electronic excitation) becomes comparable or even larger than the corresponding electronic transition frequency. In such regime, it is possible to manipulate the quantum ground state of the cavity system, to modify the decoherence properties of the system [10] and to enhance interesting non-adiabatic cavity QED effects [4,11].

Recently, it was theoretically predicted that by coupling a cavity photon mode to the cyclotron transition (of frequency ω 0 ) of a two-dimensional electron gas (2DEG) in a semiconductor, it is possible to have a vacuum Rabi frequency Ω 0 such that Ω 0 /ω 0 ∼ √ α fs ν n QW where α fs is the fine structure constant, ν is the filling factor of the Landau levels and n QW is the (effective) number of quantum wells [12]. Such a predicted scaling has been quantitatively demonstrated by recent impressive spectroscopy experimental results in the THz domain [13] .The striking consequence of this physical behavior is that for high filling factors it is possible to have Ω 0 /ω 0 ≫ 1. In this ultrastrong coupling limit for the 2DEG, however, no instability is expected for the ground state, as a result of the role played by the so-called diamagnetic Â2 -term [14] ( Â is the electromagnetic vector potential operator). In the case of massive electrons in a semiconductor, the effective mass approximation is known to work very well in a broad range of conditions and one can generally consider just the conduction band to describe the physics of the two-dimensional electron gas, hence the underlying crystal structure of the semiconductor host turns out to be unimportant for many physical effects. In the case of graphene with massless fermions [15][16][17], this is certainly not the case. Some recent experimental works [16,18,19] have demonstrated a relatively high carrier mobility in different graphene devices in the high density regime ( 10 4 cm 2 V -1 s -1 ) leading a well defined cyclotron resonance even in the case of small magnetic fields. A clearly intriguing problem is to explore how graphene behaves when embedded in a cavity resonator. In particular, is it possible to achieve ultrastrong coupling between a cavity photon and the cyclotron transition in graphene ? If yes, is there any qualitative difference with respect to the case of massive fermions in semiconductors ?

In this letter, we present a microscopic theory for the cavity QED of graphene under perpendicular magnetic field, giving an answer to the questions formulated above. Through a quantum field approach (accounting also for Coulomb depolarization effects) we show that the ultrastrong coupling regime is achievable for graphene. Moreover, due to the emergence of the Dirac cones with linear energy dispersions, the role of the Â2 turns out to be negligible when the lattice constant is much smaller than the photon wavelength. We predict the occurrence of a vacuum instability, which is absent in the case of the 2DEG of massive electrons.

Graphene is a single layer of carbon atoms forming a honeycomb lattice with two equivalent triangular Bravais sublattices A and B. Each atom of A (B) type is connected to its 3 nearest neighbors of B (A) type via the displacement vectors e 1 = a √ 3/2u x + a/2u y , e 2 = -a √ 3/2u x + a/2u y and e 3 = -au y . We consider here the tight-binding description taking only the first nearest neighbors into account with a hopping parameter t. In this case, the first quantization Hamiltonian in Q-space can be expressed in a two-components spinor basis due to the sublattice isospin [20], namely:

H Q = 0 h Q h * Q 0 ( 1
)
where h Q = -t 3 j=1 e -iQ•ej . In the vicinity of the two inequivalent Dirac points (valleys) K and K ′ marked by the two vectors K ± = ±4π/(3 √ 3a)u x , the lowenergy excitations are well described by the massless Dirac Hamiltonian H α,κ = v F (ακ x σ x + κ y σ y ) where κ = Q -K ± , σ i (i = x, y, z) are the Pauli matrices, v F = (3at)/(2 ) is the Fermi velocity (∼ 10 6 m/s) and α = ± is the valley isospin index.

Here, we are interested in the case where a static magnetic field B is applied perpendicularly to the graphene plane. This problem can be exactly solved by means of the Peierls substitution (which remains valid as long as a is much smaller than the magnetic length l 0 = ( c)/(eB) in the Dirac Hamiltonian), i.e. replacing κ with Π 0 = p + e c A 0 where A 0 = -Byu x is the vector potential (Landau gauge). This yields the graphene Landau Levels [21] whose energies are

E λ=± n = λ ω 0 √ n
where n ≥ 0, λ refers to the electron (λ = +) or hole band (λ = -) and

ω 0 = v F √ 2/l 0 .
Each Landau level has a degeneracy N = g s S/(2πl 2 0 ), S being the surface of the graphene layer and g s = 4 accounts for the spin and valley isospin degeneracy (we neglect Zeeman and valley splittings). Finally we define the filling factor (we use a different convention than the usual one [16,17]) as ν = ρS/N + 1/2 (ρ is the total electron density) so that ν = n corresponds to the situation where the last fully occupied Landau level has an orbital quantum number n -1. In the following, we will study the case of integer filling factors and consider for simplicity the case of Fermi level in the electron band (λ = +). Moreover, we will consider a graphene layer embedded in a 0D cavity resonator. We investigate the coupling between a particular cavity photon mode and the transition between the last occupied Landau level n = ν -1 and the first unoccupied one n = ν with a frequency

ω 0 ∆ ν = ω 0 √ ν - √ ν -1 .
The Dirac fermion field operator in the valley α is written as a two-components spinor

Ψα (r) = λ,n,k    ψ (A) λ,n,k,α (r) ψ (B) λ,n,k,α (r)    c λ,n,k,α , (2)
where the wavefunctions in the valley K are

   ψ (A) λ,n,k,+ (r) ψ (B) λ,n,k,+ (r)    = e ikx √ L   -λC - n ϕ n-1,k (y) C + n ϕ n,k (y)   δ z - L z 2 .
(3) The operator c λ,n,k,α annihilates a fermion in the state labelled by the quantum numbers (λ, n, k, α). C - n = 1δ n,0 /2 and C + n = 1 + δ n,0 /2 are the normalization factors, L is the length of the graphene sheet (taken squared for simplicity) along the x and y directions, L z is the cavity length along the z direction and ϕ n,k (y) corresponds to the eigenfunction of the 1D harmonic oscillator problem shifted by the guiding centre position y 0 = kl 2 0 . The wavefunctions in the valley K ′ are obtained by swapping the spinor (3) and changing λ → -λ.

We consider a 0D cavity resonator (see Fig. 1) of volume V = L z L 2 that confines the electromagnetic modes along the three spatial directions. These cavity modes are labelled by the quantized wavevector q ≡ (q x , q y , q z ) ≡ (πn x /L, πn y /L, πn z /L z ), where n x , n y , n z are integer numbers. For sake of simplicity, we will consider that the cavity length L z along the z direction is much smaller than the cavity transverse size L. This way, we can restrict our study to the particular photon mode with n z = 1, neglecting all the higher-lying modes (n z > 1). We will also assume that the graphene layer is placed in the middle of the cavity at z = L z /2 (see Fig. 1). Of course, our theory can be easily generalized to a more complicated geometry. The electromagnetic vector potential reads [22]

Âem (r) = q,η 2π c 2 ǫ ω cav (q)V a q,η u q,η + a † q,η u * q,η (4)
where η = 1, 2 is the photon polarization, ω cav (q) is the cavity frequency, ǫ the cavity dielectric constant and [START_REF]= 2 if ni = 0 and Cq i = 2 √ 2 else[END_REF] u q,1 =   C qx cos(q x x) sin(q y y) sin(πz/L z ) cos θ cos φ C qy sin(q x x) cos(q y y) sin(πz/L z ) cos θ sin φ -2 √ 2 sin(q x x) sin(q y y) cos(πz/L z ) sin θ

  , (5)
u q,2 =   -C qx cos(q x x) sin(q y y) sin(πz/L z ) sin φ C qy sin(q x x) cos(q y y) sin(πz/L z ) cos φ 0   , (6)
with cos θ = q z /|q| and cos φ = q x / q 2 x + q 2 y . The operator a q,η is the photon annihilation operator in the cavity mode (q, η).

When considering the graphene sheet in the presence of a perpendicular static magnetic field embedded in our cavity resonator, the electron velocity operator involves the total vector potential ÂT = A 0 + Âem . The coupling Hamiltonian is obtained by replacing the momentum p in the free Hamiltonian by its gauge invariant form Π = p + e c A T = Π 0 + e c A em . The Hamiltonian (1) becomes

h α = -t 3 j=1 Z (α) j exp - i Π 0 • e j + e c A em • e j , (7)
where we have defined the valley-dependent phase factor Z (α) j = e -iK ± •ej . As we are interested in the quantum ground state properties and low-energy excitations, we can consider the continuum limit (lattice size a → 0, which is the relevant limit when a much smaller than any other length scale). In other words, we expand the Hamiltonian (7) with respect to the lattice parameter a and retain the leading contribution. This way, we get two terms, the first one is the massless Dirac Hamiltonian H (α) 0 with static magnetic field, while the second one

H (α)

int represents the coupling between the Dirac fermions and the cavity optical modes:

H (α) 0 = v F (αΠ 0,x σ x + Π 0,y σ y ) (8)
H (α) int = v F e c (αA em,x σ x + A em,y σ y ) (9)
Note that the coupling Hamiltonian ( 9) depends lin-early on the bosonic operators a q,η and a † q,η . Consequently, the total Hamiltonian does not contain any term involving the squared vector potential A 2 em as it is usually the case in cavity QED for massive electrons. Using Eqs. ( 2), ( 4) and ( 9), the second quantized coupling Hamiltonian H int = d 2 r Ψ † (r)H (α) int Ψ(r) becomes valley independent so we can omit the index α taking the valley degeneracy into account through the factor g s . Note that since we are dealing with optical modes, we have |q|l 0 ≪ 1. This condition allows us to neglect the LL mixing [24] when considering only the transition ν → ν -1. The final result is

H int = q,η Ω q,η a q,η + a † q,η d q,η + d † q,η (10)
where Ω q,1 = -Ω q cos θ, Ω q,2 = Ω q with the vacuum Rabi frequency (for ν > 1) given by

Ω q = ω 0 α fs g s 4π √ ǫ . ( 11
)
The boson annihilation operators d q,η (polarization η = 1, 2) corresponding to the bright collective modes are:

d q,1 = 1 √ N k sin q y k + q x 2 l 2 0 -φ c † ν-1,k c ν,k+qx + 1 √ N k sin q y k - q x 2 l 2 0 + φ c † ν-1,k c ν,k-qx , ( 12
)
d q,2 = 1 √ N k cos q y k + q x 2 l 2 0 -φ c † ν-1,k c ν,k+qx - 1 √ N k cos q y k - q x 2 l 2 0 + φ c † ν-1,k c ν,k-qx . ( 13
)
The first relevant conclusion of our letter is that graphene can enter deeply the ultrastrong coupling regime. In fact, in the high filling factor regime (i.e. ν ≫ 1) the cyclotron transition frequency is ω 0 ∆ ν ∼ ω 0 /(2 √ ν). Hence, the vacuum Rabi frequency normalized to the transition frequency

Ω q ω 0 ∆ ν ≃ √ ν α fs g s π √ ǫ ( 14
)
can be larger than 1 for large enough filling factors. To complete our treatment, we consider the Coulomb interaction following a bosonization procedure [25][26][27] allowing us to calculate the depolarization shift (Random phase approximation (RPA) contribution). The second quantized Coulomb Hamiltonian reads:

H coul = 1 2 d 2 r d 2 r ′ Ψ † (r) Ψ(r) e 2 ǫ|r -r ′ | Ψ † (r ′ ) Ψ(r ′ ).
(15) In the considered geometry, it is convenient to expand the Coulomb potential V (rr ′ ) using a 2D Fourier series. Finally, we use the same procedure as for the lightmatter coupling Hamiltonian derivation restricting the Coulomb Hamiltonian to describe only scattering processes between LLs ν -1 and ν thanks to the condition |q|l 0 ≪ 1. This provides the Coulomb Hamiltonian written in terms of the collective modes d q,η and e q,η

H coul = q,η V q γ η d † q,η + γ η d q,η 2 -e † q,η -γ η e q,η 2 , (16)
with γ 1 = -1, γ 2 = 1 and where V q = α fs gsc|q ⊥ | 32ǫ ξ 2 ν , q ⊥ = (q x , q y ) denotes the in-plane wavevector and ξ ν =

C - ν-1 √ ν -1 + C + ν-1 √ ν.
In the high filling factor regime, we have ξ ν ∼ √ 2ν. The Coulomb Hamiltonian reveals the presence of two additional dark modes (i.e uncoupled to the electromagnetic field) corresponding to two operators e q,η and e † q,η . Since the bright and dark modes are decoupled by the Coulomb interaction, if we look at the ground state properties, we can omit the dark modes. After bosonizing the kinetic energy [25,26], the Hamiltonian reads:

H dp = q,η ω 0 ∆ ν d † q,η d q,η + V q γ η d † q,η + γ η d q,η 2 = q,η ω p (q) m † q,η m q,η + const. ( 17
)
where m q,1 and m q,2 corresponds to the magnetoplasmon modes with energy ω p (q) = ω 0 ∆ ν (ω 0 ∆ ν + 4V q ). Since our bosonic approach allows us to calculate the RPA contribution [25], we find that V q → 0 when q → 0. However, our treatment does not include the vertex corrections (attractive electron-hole Coulomb interaction ) nor the difference between the exchange self-energy of the electron and that of the hole. For this reason, we do not find the renormalization of the transition frequency ω 0 ∆ ν at zero wavevector which is expected since the Kohn's theorem [28] does not apply in graphene [24]. Nevertheless, some recent theoretical studies [29][30][31] have shown that these corrections lead to a relatively moderate renormalization (< 20%) of the Fermi velocity and can thus be easily accounted for in our model. In terms of the magnetoplasmon modes, the Hamiltonian reads:

H tot = q,η ω p (q) m †
q,η m q,η + ω cav (q)a † q,η a q,η + Λ q,η a q,η + a † q,η

m q,η + m † q,η (18)
where Λ q,1 = Ω q cos θ ωp(q) ω0∆ν and Λ q,2 = Ω q ω0∆ν ωp(q) . In contrast to the Hopfield [32] quantum Hamiltonian for massive quasi-particles in semiconductor microcavities [START_REF] Ciuti | [END_REF]5], such an Hamiltonian is reminiscent of the Dicke Hamiltonian, which admits a quantum critical point (QCP) beyond which the normal ground state becomes unstable. Diagonalizing the Hamiltonian (18) via the Bogoliubov method, we find that the energy of the lower eigenvalue vanishes (existence of a gapless excitation) when Ω q = ω0∆ν ωcav(q) 4 for the branch η = 1

and Ω q = ω 2 p (q)ωcav(q) 4ω0∆ν

for the branch η = 2. Note that the no-go theorem for cavity QED quantum phase transitions [14] holds as long as the effective kinetic energy takes the usual quadratic form p 2 /2m * which well describes massive electrons in semiconductors (m * is the electron band effective mass). In graphene, the peculiar crystal structure is essential in determining the electronic properties and the effective kinetic energy is governed by the massless Dirac Hamiltonian [15,20] v F p • σ at low energy (σ = (σ x , σ y ) being the Pauli matrices). Since this Hamiltonian is linear in p, such no-go theorem does not necessarily apply and indeed our work (which starts from the tight-binding Hamiltonian of graphene) predicts that a vacuum instability occurs.

ω j,η /ω cav (q) ω UP,2 ω UP,1 ω LP,2 f p (q) ω cav (q) f UP,1 f cav (q) ω LP,1 f LP,1 f LP,2 f UP,2 ω p (q)
FIG. 2: Left panel: normalized frequencies ωj,η/ωcav(q) (solid lines) of the graphene magnetopolariton lower and upper branches as a function of the carrier density for a given magnetic field B = 50 mT. The dashed lines depict the bare cavity and magnetoplasmon frequencies. Parameters: ǫ = 3.9, Lz = 760 µm and L = 5Lz, optical mode with q ≡ (nx = 1, ny = 1, nz = 1) and ωq = 0.628 THz rad -1 (fq = 0.1 THz). Right panel: same quantities (unnormalized and divided by 2π) as a function of the magnetic field just below the critical density ρc = 1.4 × 10 12 cm -2 .

The left panel of the Fig. 2 shows the frequencies of the magnetopolariton excitations ω j,η , (j = LP, U P ) normalized to the cavity mode frequency versus electron density. We see that by increasing the density, the lower branch η = 1 has a nonmonotonical behavior and vanishes when reaching a critical value [START_REF]Considering the high filling factor limit (ν ≫ 1)[END_REF] ρ c = 1.4×10 12 cm -2 . On the right panel, the magnetopolariton frequencies f j,η = ω j,η /2π are depicted as a function of the magnetic field just below the critical density ρ c . We notice a strongly asymmetric dispersion when increasing the magnetic field together with the vanishing frequency of the lower branch f LP,1 . These characteristics are a signature of such a vacuum instability. In order to describe the properties of the Dicke Hamiltonian above the QCP, an Holstein-Primakoff [START_REF] Holstein | [END_REF] bosonization procedure [35] can be applied. However, such a mapping does not hold in this case because of the non-vertical excitations at finite wavevector [26], making the analytical description of the excitations above the QCP highly non-trivial and will be the scope of future investigations.

In conclusion, we have shown that the peculiar crystal structure of graphene leads to remarkable cavity QED properties. In particular, the coupling of the graphene cyclotron transition is qualitatively different for the case of massive electrons in semiconductors. In the ultrastrong coupling regime, a vacuum instability analogous to the one occuring in the Dicke-model can also occur for graphene, whereas for massive quasi-particles this is not the case. In free space graphene (without cavity), the anomalous properties of Dirac quasiparticles have been shown to lead to an unusual behavior of the optical conductivities in the presence of a perpendicular magnetic field [36]. For cavity embedded Dirac fermions, our work paves the way to interesting developments, such as cavity-controlled magnetotransport in graphene. We would like to thank M. O. Goerbig, S. De Liberato and P. Nataf for fruitful discussions.

FIG. 1 :

 1 FIG. 1: Left panel: sketch of a cavity resonator of volume V = LzL 2 (Lz ≪ L) embedding a graphene layer with an uniform and static magnetic field B perpendicular to the atomic plane. Right panel: a fully confined cavity photon mode is supposed to be quasi-resonant to the graphene cyclotron transition between LLs n = ν -1 and n = ν (red horizontal solid lines).