Quantum state measurement Introduction The rotation properties of light fields of spin are described by the 3 3 representation of the 0 0 SO(3) group, with the generators JJ ii we found in class, for instance JJ zz = 0 0 0. Then, 0 0 ee iiii 0 0 the rotation operator for the same zz axis is RR zz (ββ) = 0 0. It would appear that 0 0 ee iiii the light field, having spin, would require a basis set made of three states. However, the field component along the beam momentum is AA aaaaaaaaaa kk = 0 for the radiation electromagnetic fields (the most common case, in the far-field region, away from the sources) and two states are sufficient for the basis set. It is possible to describe the rotation properties of these transverse light states in a basis set made of only two kets with mm = ±, rather than three, because the mm = 0 component is not present. There are many possible two-state basis sets, but we will use the one made of the HH, VV states (horizontally- and vertically-polarized light), because it maps well to a method of calculating the polarization of light in classical optics, called the Jones calculus (Appendix). In quantum mechanics, light states αα are written in this basis set as αα = aa HH + bbee iiii VV or represented by αα aa bbee iiii, where aa, bb are real numbers. We set aa 2 + bb 2 = to have normalized states, as usual. Since optical elements, for instance QWP or HWP phase retarders, vary the polarization of light, they must be described by operators, acting on these quantum states of light. It turns out that we can simply take the matrices describing these optical elements in classical Jones calculus as the representations of operators HHHHHH, QQQQQQ in the HH, VV basis set. Specifically, for a half-wave- and quarter-wave-plate with the fast axis at an angle φφ from the horizontal, HHHHHH cos 2φφ sin 2φφ (φφ) and sin 2φφ cos2 φφ QQQQQQ (φφ) (cos φφ)2 + ii(sin φφ) 2 ( ii) cos φφ sin φφ ( ii) cos φφ sin φφ ii(cos φφ) 2 + (sin φφ) 2, respectively. Example: a QWP is followed by a HWP in the light beam. We can find the final state αα by calculating αα = HHHHHH (φφ 2 )QQQQQQ (φφ )(aa HH + bbee iiii VV ) and confirm that its projection along the horizontal axis is given in the three following cases by: Page
φφ = 0, φφ 2 = 0. Then, αα aa iiiiee iiii and HH αα 2 = aa 2 φφ = ππ, φφ 4 2 = 0. Then, αα + ii ii aa 2 + ii ii bbee iiii and HH αα 2 = aa( + ii) + 2 bbeeiiii ( ii) 2 = ( + 2aaaa sin φφ) 2 φφ = ππ, φφ 4 2 = ππ. Then, 8 αα aa ii ii bbee iiii and HH αα 2 = aa + bbeeiiii 2 = ( + 2aaaa cos φφ) 2 Note: φφ corresponds to the QQWWWW, φφ 2 corresponds to the HHWWWW. These measurements are useful if we want to find the initial quantum state of light αα. We require three measurements, since we need two real numbers aa, bb and one phase φφ. The three measurements above are necessary and sufficient (the condition aa 2 + bb 2 = is applied to normalize the intensity measurements). 2 Setup We will use the setup with the AA, BB, BB detectors (we do not need the AA detector) The polarizing beam splitter (PBS) in the BB beam lets the pp polarization through to BB, while the ss polarization component is reflected to BB Post the door with the warning sign. Turn on the 405 nm laser and add the BBO crystal to its post Turn on the FPGA card Turn on the detectors for AA, BB, BB As usual with these measurements, the overhead light and other sources (phones, flashlights, etc.) must be off at all times. Start the vi called QSM_rs232_2.6 (it will use the Lab View version 20) Add the 80 nm HWP in the infrared BB beam. Find the zero orientation of HWP (when the fast axis is horizontal), by finding its rotation angle that maximizes AAAA. Enter this number in the vi window. Similarly, add the 80 QWP and find its zero orientation (fast axis horizontal) when AAAA is maximum. Enter this number in the vi window. The detections in the BB, BB channels are conditioned by the detection in the AA channel and therefore we are looking at a single-photon beam. The combination of QWP, HWP, PBS and the two detectors BB, BB form an apparatus that can be applied to measuring quantum states of different polarizations for the single-photon infrared BB beam. Page 2
3 Experiments To vary the light beam quantum state, additional polarization elements between the crystal and the HWP, QWP you aligned must be inserted We call these new elements QQQQPP gg, HHHHPP gg for state generator-qwp, and generator- HWP Add the HHHHPP gg (no QQQQPP gg inserted). How should we rotate this to obtain the + 45 state after it, with the polarization of the single-photon beam rotated by +45 degrees (as looking along the beam direction) from the horizontal? Classical Jones calculus works fine here. Measure the aa, bb, φφ parameters of this state by doing the set of three measurements from the introduction and compare to the expected result. Replace the HHHHPP gg with the QQQQPP gg and generate the left-circularly polarized state LL. How should we orient the QWP (classical Jones calculus work well here)? Measure the aa, bb, φφ parameters of the state by doing the set of three measurements and compare to expectations. Turn off the detectors, the 405 nm laser, and return the BBO crystal to the jar. 4 Conclusion The measurements in this experiment can be described with the classical Jones calculus. Why do we do these complicated polarization measurements of quantum states and call a matrix already well-known in classical optics, a representation of an operator? These measurements will be useful in a famous experiment (Hardy s test ) that allowed an experimental verification of one astonishing consequence of quantum mechanics postulates. 5 Appendix Light and optical elements in Jones calculus of classical optics A matrix formalism is introduced in polarization calculations of classical optics, to speed up calculations. Classical polarization states of light are replaced with 2-element columns, with the two elements corresponding to the horizontal and vertical component of the light, respectively. Each optical element is also replaced with a 2 2 matrix. The advantage of this method is that Page 3
the action of a series of optical elements on a light state can be replaced by the matrix obtained by multiplying the matrices corresponding to individual elements (the ``Jones calculus ). For instance, a HWP with the fast axis horizontal (this would be the zero of the angle θθ) is replaced with the matrix HWP(0) = 0. As you can see what this matrix does to a light 0 state h vv is to reverse the sign of the vv component, equivalent to multiplication by eeiiii. In other words, the vertical component has been delayed (fast axis horizontal) by λλ (half-wave plate) or 2 ii2ππ λλ 2 by a phase ee λλ = ee iiii. Similarly, a QWP with fast axis horizontal has a matrix QWP(0) = 0. There is no quantum mechanics here, just mathematical convenience. 0 ii To find the matrices corresponding to optical elements rotated in an orientation with the fast axis at an arbitrary angle, we can apply the classical rotation transformation. For instance, cos φφ sin φφ cos φφ sin φφ 2φφ sin 2φφ HHHHHH(φφ) = HWP(0) = cos sin φφ cos φφ sin φφ cos φφ sin 2φφ cos2 φφ. When θθ = ππ 2 (fast axis vertical) we get HHHHHH ππ 0 = 2 0 as expected. When θθ = ππ we get HHHHHH 4 ππ = 4 0 0, or when acting on a horizontal state 0 we get 0, as expected. Similarly, cos φφ sin φφ cos φφ sin φφ QQQQQQ(φφ) = QWP(0) sin φφ cos φφ sin φφ cos φφ = (cos φφ)2 + ii(sin φφ) 2 ( ii) cos φφ sin φφ ( ii) cos φφ sin φφ ii(cos φφ) 2 + (sin φφ) 2. Light in quantum mechanics Following the qualitative arguments in Sakurai Ch., RCP and LLLLLL = ii HH ± the RCP/LCP basis set HH are complete and orthogonal. Page 4 and VV ii ii VV. Then, in. Both { HH, VV } and { LL, RR } basis sets Example: the representation of spin angular momentum in the LL, RR basis set is JJ zz 0 0. The representation of the rotation operator about the zz axis is then DD zz(rr) = ee iijj zzφφ = iiiijj zz + 2 ( ii)2 JJ 2 zz + = cos φφ ii sin φφ JJ zz ee iiii 0 (compare to the iiii 0 ee expression in the Introduction section). To find the representation of this operator in the HH, VV basis set from its representation in the LL, RR basis set, apply DD = UU + DDDD
2 ii ii ee iiii 0 ii φφ sin φφ 0 eeiiii = cos. The result is consistent with ii sin φφ cos φφ expectations from the ``active rotations (see Sakurai [3..3]). Exercise: verify that the representations of a phase retarder in the HH, VV basis set and the classical Jones calculus give the same answer in one specific case. Page 5
Name Phys-602 Quantum Mechanics Laboratory Quantum state measurement lab report Date of measurements: Page 6