Euler angles are almost never actually used for equations of motion or interpolation. They're terrible at it because they behave borderline non-physical. For instance, if you try to rotate smoothly from one orientation to another just by linearly interpolating the yaw pitch and roll channels, you find the object takes a rather exotic looking path to get to its final orientation. They're convenient for a quick way to describe an orientation in a physically meaningful way, but they act squirely when you start to vary them over time.

For our specific transitions in \(^{133}\)Cs, \(6^2S_{1/2}, F=3, m_F=0 \rightarrow 7^2P_{1/2}, F=3,4, m_F=1\) and \(7^2P_{1/2}, F=3,4, m_F=1 \rightarrow 6^2S_{1/2}, F=4, m_F=0\), the relevant quantum numbers are the initial and final total electron angular momentum quantum numbers \(J_1=J'=1/2\), the nuclear spin quantum number \(I=7/2\), the initial magnetic quantum number \(m_{F1}=0\), and we use circularly polarized Raman laser beams such that the z-components of the angular momentum of the absorbed and emitted photons are \(q_1=q_2=1\). With these, we find

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, A. Rauschenbeutel, D. Meschede, Phys. Rev. A 72, 023406 (2005)

For modern cameras, a circular polarizer (CPL) is typically used, which has a linear polarizer that performs the artistic function just described, followed by a ...

The key to working with them is to understand that they are actually describing an orientation. The individual values themselves obey strange rules which are borderline non-physical. For example, in your case where you are yaw-ing CW, and then roll 180 degrees, in the real world your orientation will keep changing the same way it did before, but the numeric value for yaw will have flipped its sign. This is okay. It obeyed the conservation laws, even though yaw could be discontinuous in its sudden jump!

F. Nogrette, H. Labuhn, S. Ravets, D. Barredo, L. Béguin, A. Vernier, T. Lahaye, A. Browaeys, Phys. Rev. X 4, 021034 (2014)

Gaussianbeam waist

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The reduced dipole matrix element for the \(6^2S_{1/2}\rightarrow 7^2P_{1/2}\) transition in \(^{133}\)Cs is \(e\langle 7^2P_{1/2}||r||6^2S_{1/2}\rangle =0.276 ea_0\) [19], where \(a_0\) is the Bohr radius.

To find the total Stark shift of each of the hyperfine ground states, we need to add the Stark shifts due to the first and second Raman laser beams, so

Gaussianbeam pdf

Restating my question again, is it realistic for a 180 degree roll to seemingly reverse the rotation direction of a yaw movement?

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Gaussianbeam intensity formula

Also, a shout out to screw theory. While a spaceship simulator tends to use positions and quaternions, screw theory combines the two into one 6 dimensional concept: the screw. This is popular in robotics because you have 6 dimensions of freedom and a 6-element vector. This makes it easier to do "inverse kinematics" where you try to achieve a particular positioning of a tool using all of the joints of a robotic arm. It's easier because there's natural concepts of "inverting" a joint rotation which are less natural when you break things up into positions and orientations.

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I was really intrigued by this so I went to look for a physical example. I took a skateboard wheel and spun it so that it was spinning CCW, then flipped the board over, and indeed the wheel was now spinning CW to me. Still, this seems really suspicious to me, and I'm worried there may be a difference between the wheel and a spaceship, given that the first one is fixed on an axis, and the second one is just freely rotating in space.

For each hyperfine ground state, we must sum over the contributions due to each of the hyperfine states of the \(7^2P_{1/2}\) manifold, resulting in

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, D. Meschede, Phys. Rev. Lett. 93, 150501 (2004)

Or more generally, is the case where the movement makes intuitive sense to the pilot the correct one? Or should it follow the intuition of the outside perspective?

(I'm not a physics person, so please don't be too hard on me if I've used some terminology wrong, hopefully my explanation makes my question clear even if the terminology is wrong)

Gaussianbeam profile

104:59:35 Garriott: Columbia, Houston. We noticed you are maneuvering very close to gimbal lock. I suggest you move back away. Over. 104:59:43 Collins: Yeah. I am going around it, doing a CMC Auto maneuver to the Pad values of roll 270, pitch 101, yaw 45. 104:59:52 Garriott: Roger, Columbia. (Long Pause) 105:00:30 Collins: (Faint, joking) How about sending me a fourth gimbal for Christmas.

Roll Pitch and Yaw are common names for 3 angles used to describe orientation. There's actually a lot of ways they can combine. At the top level you have Euler Angles vs Tait-Bryan Angles, which refer to which axes you rotate around at a very high level (xyx vs. xyz). The confusion starts right there, as most aircraft use yaw-pitch-roll as Tait-Bryan angles, but people will use the name "euler angles" to describe them! Beyond these 2 categories, there are dozens to over a thousand of ways they can be combined depending on how you phrase the question and which things you are willing to vary (in particular, whether you are willing to change the order of the 3 axes or not and whether you allow the axes to point in the opposite direction).

K. Maller, M.T. Lichtman, T. Xia, Y. Sun, M.J. Piotrowicz, A.W. Carr, L. Isenhower, M. Saffman, Phys. Rev. A 92, 022336 (2015)

Definition of Absorption of Light. What is Absorption of Light? Absorption of light takes place when matter captures electromagnetic radiation, converting the ...

The Rabi oscillations investigated in this work are driven via a Raman process from the \(F=3, m_F=0\) hyperfine ground state of the \(6^2S_{1/2}\) manifold in \(^{133}\)Cs to its \(F=4,m_F=0\) hyperfine ground state via the \(7^2P_{1/2}\) manifold using two laser beams. To treat this kind of Rabi oscillation, we repeat the steps from Sect. 2.4 for a \(\varLambda\)-type three-level system with two lasers tuned to the two transitions of the Raman process. For detunings large enough so that the excited-state population is small, we can adiabatically eliminate the \(7^2P_{1/2}\) state, resulting in an effective two-level Rabi oscillation with an on-resonance Rabi frequency of

Taking into account the hyperfine splitting of the \(7^2P_{1/2}\) level, we have to sum over all possible intermediate states, resulting in

The one-photon Rabi frequencies are \(\varOmega _{i,F_i F_f}=\varOmega _{i,0} \tilde{\varOmega }_{F_i F_f}\) with \(\varOmega _{i,0}= {\mathcal {E}}_i e \langle 7^2P_{1/2}||r||6^2S_{1/2}\rangle /\hbar\), where \({\mathcal {E}}_i\) is the electric field amplitude of Raman laser \(i=1,2\), \(e\langle 7^2P_{1/2}||r||6^2S_{1/2}\rangle\) is the reduced dipole matrix element for the \(6^2S_{1/2} \rightarrow 7^2P_{1/2}\) transition, e is the elementary charge, and

Gillen-Christandl, K., Gillen, G.D., Piotrowicz, M.J. et al. Comparison of Gaussian and super Gaussian laser beams for addressing atomic qubits. Appl. Phys. B 122, 131 (2016). https://doi.org/10.1007/s00340-016-6407-y

where we used primes to indicate quantum numbers pertaining to the excited states. The detunings from the \(7^2P_{1/2}\) hyperfine states are \(\varDelta _{R,3'}=\varDelta _R-\varDelta _{F'3}\) and \(\varDelta _{R,4'}=\varDelta _R-\varDelta _{F'4}\). Here, \(\varDelta _{F'3}=-2\pi \times 212.3\,\text {MHz}\) and \(\varDelta _{F'4}=2\pi \times 165.1\,\text {MHz}\) are the hyperfine shifts from the \(7^2P_{1/2}\) fine structure level to the \(F'=3,4\) hyperfine states, respectively.

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There is a 4 ring gyro design which is immune to these issues. However, at the time, the development of the hardware would have pushed the schedule too hard, so they stuck with the simpler 3 ring gyro. Pilot Mike Collins had some choice words about this in flight:

If you want a sense of how wonky euler angles can be, look at gimbal lock. This is a quirky situation where two of the axes generate the same rotation. It happens when the second rotation rotates the third rotation into the same axis as the first rotation. This is a major issue with gimbals because, when it occurs, you end up with one rotational dimension that you simply cannot observe. This means that once gimbal lock occurs, you no longer know what your orientation is.

In \(^{133}\)Cs, we have \(F_1=3\), \(F_2=4\), and \(F'=3,4\). We thus find for the two-photon on-resonance Rabi frequency

Gaussian laserbefore and after

for a stimulated Raman emission. Here, \(C_{F_1,m_{F1},1,q_1}^{F',m_{F1}+q_1}\) and \(C_{F',m_{F1}+q_1,1,-q_2}^{F_2,m_{F1}+q_1-q_2}\) are Clebsch–Gordan coefficients, and

ABCD matrixGaussianbeam

T. Xia, M. Lichtman, K. Maller, A.W. Carr, M.J. Piotrowicz, L. Isenhower, M. Saffman, Phys. Rev. Lett. 114, 100503 (2015)

D.D. Yavuz, P.B. Kulatunga, E. Urban, T.A. Johnson, N. Proite, T. Henage, T.G. Walker, M. Saffman, Phys. Rev. Lett. 96, 063001 (2006)

If you want to have things like a rate of change of orientation, use a orientation format which supports that better. Direction Cosine Matrixes and Quaternions have far superior properties. From my experience, quaternions are by far the most popular way to handle rotations, although it takes a while to wrap your head around them. Both of them have straightforward ways of dealing with the conservation of angular momentum.

where \(\varOmega _{1,2}\) are the single-photon on-resonance Rabi frequencies for the \(6^2S_{1/2}, F=3, m_F=0 \rightarrow 7^2P_{1/2}\) and \(7^2P_{1/2} \rightarrow 6^2S_{1/2}, F=4, m_F=0\) transitions, respectively. \(\varDelta _R\) is the detuning of the first Raman laser beam from the \(6^2S_{1/2}, F=3 \rightarrow 7^2P_{1/2}\) (fine structure level) transition, and we have assumed that the detuning of the second Raman laser from the \(6^2S_{1/2}, F=4 \rightarrow 7^2P_{1/2}\) transition is the same. Equation (20) is valid for two-photon resonance or when the departure from two-photon resonance is small compared to \(\varDelta _R\).

where \(F', F_{1,2}\) are the total angular momentum quantum numbers of the intermediate, initial, and final states, respectively, \(\varOmega _{1,F_1 F'}\) and \(\varOmega _{2,F' F_2}\) are the single-photon on-resonance Rabi frequencies for the \(F_1 \rightarrow F'\) and \(F' \rightarrow F_2\) transitions, respectively, and \(\varDelta _{R,F'}\) is the detuning of the first Raman laser beam from the \(F_1 \rightarrow F'\) transition.

E. Mount, C. Kabytayev, S. Crain, R. Harper, S.Y. Baek, G. Vrijsen, S.T. Flammia, K.R. Brown, P. Maunz, J. Kim, Phys. Rev. A 92, 060301(R) (2015)

2015721 — A microscope is an optical instrument used to view objects that are too small to be seen clearly by the naked eye. It magnifies the image of ...

Gaussianbeam calculator

We study the fidelity of single-qubit quantum gates performed with two-frequency laser fields that have a Gaussian or super Gaussian spatial mode. Numerical simulations are used to account for imperfections arising from atomic motion in an optical trap, spatially varying Stark shifts of the trapping and control beams, and transverse and axial misalignment of the control beams. Numerical results that account for the three-dimensional distribution of control light show that a super Gaussian mode with intensity \(I\sim \hbox {e}^{-2(r/w_0)^n}\) provides reduced sensitivity to atomic motion and beam misalignment. Choosing a super Gaussian with \(n=6\) the decay time of finite temperature Rabi oscillations can be increased by a factor of 60 compared to an \(n=2\) Gaussian beam, while reducing crosstalk to neighboring qubit sites.

Gaussianbeam equation

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For example, take the following scenario. From the perspective of the pilot, I'm yawing left at some angular speed x. If I look at the ship from outside in a "top down" (top relative to the pilot), I would see the ship rotating CCW with angular speed x. Then, let's say I roll clockwise 180 degrees from the perspective of the pilot. Here's where things get a bit trippy to me. My intuition, and what happens in game, both produce the following result: From the pilot's perspective, the ship is still yawing left at the same speed x. However, from the "top down" perspective, the ship would appear to be rotating CW with speed x. So, a 180 degree roll changed the ships rotation from CCW to CW. Does that make sense? Is angular momentum conserved here? So in a sense the way I've implemented it makes intuitive sense to the pilot (if I'm yawing left, and I do some rotation in another axis, my yaw shouldn't be affected so I should still be yawing left). However, to an "outside" perspective, this makes it look as if the ship is just sometimes randomly reversing rotational direction. On the other hand, if I make it intuitive for the outside perspective, in the earlier example I gave, the ship would instead be yawing right for the pilot after the roll maneuver, which would make it continue rotating CCW; making more sense to the "top down" view, but being trippier for the pilot.

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In this work, we used two Raman laser beams of identical power, waist, and alignment, so \({\mathcal {E}}_1={\mathcal {E}}_2\), and consequently

We nearly failed to go to the moon because of this. Apollo 11 had a standard 3 ring gyro, so was susceptible to gimbal lock. To combat this, they had a clever little "kicker" device which would rotate one of the axes at a critical moment to sidestep this horrid alignment from occuring. As it so happened, the code to handle this had a bug, and caused the subsystem to cease operating. They flew onward knowing the risk that a gimbal lock would cause them to lose track of their attitude. (Contingency plans included re-acquiring alignment by observing stars, which would be slow and error prone, but would get astronauts home safely).

I'm making a spaceship piloting game, and I'm trying to make the movement follow the laws of physics. The ship is able to control it's angular acceleration in pitch, yaw, and roll. The way I've implemented it, the ship rotates around it's local x, y, and z axis, however, I'm having a hard time wrapping my head around rotations in 3D to verify if this kind of movement is actually accurate.

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