ensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y
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1 Ph195a lecture notes, 1/3/01 Density operators for spin- 1 ensembles So far in our iscussion of spin- 1 systems, we have restricte our attention to the case of pure states an Hamiltonian evolution. Toay we ll nee to move on to ensity operators so that we can inclue issipation an ephasing to arrive at the full Bloch Equations. We ve alreay seen that an arbitrary pure state for a two-level system can be represente by a vector on the Bloch Sphere. If we write the quantum state vector in the form è cos exp i z è sin exp i z è, then the corresponing Bloch vector connects the origin (center) of the Bloch Sphere to a surface point at polar (latitue) angle an azimuthal (longitue) angle. By convention we assign the Bloch Sphere a raius of 1, so that the Cartesian components of the Bloch vector v are simply v x sin cos, v y sin sin, v z cos. We have also note that the Bloch vector is proportional to S Ò/ è è Ò/ v, since ès x è Ò sin cos, ès y è Ò sin sin, ès z è Ò cos. When working with ensity operators, we can use this connection to efine a generalize Bloch vector: v x Tr x, v y Tr y, v z Tr z. In other wors, we set v è è Ò S for both pure states an ensity operators. It is worth noting that for a general mixe state, v â 1. For example, if 1 1 then v 0 since the Pauli matrices i 1 0 x, y, z 1 0 i are all traceless Tr 0. In cases where correspons to a known ensemble then p èè, 1
2 è è Tr Tr p èè p Tr èè èè p è è Tr èè p v, where v è è an the summation is a vector sum. In other wors, the generalize Bloch vector corresponing to an ensemble of pure states is the weighte vector sum of the Bloch vectors representing the members of the ensemble. Hence we fin that for any equally-weighte two-member ensemble with è 1 è 0, the ensemble Bloch vector will be zero since 1 è an è correspon to antipoal points. Inee, we know that 1 1 èè 1 èè 1 1 for any such ensemble. Similarly, we see that the generalize Bloch vector corresponing to 1 è x è, p 1, è x è, p, 3 è z è, p 3 1, will simply be v x,v y,v z 0,0,1. It is worth noting that any ensity operator may be expresse in the form 1 1 v, by virtue of the orthogonality of the Pauli matrices: Tr i 0, Tr i i. Hence, we fin that v is ust as general as for the purpose of specifying mixe quantum states! The generalize Bloch vector v has only three real parameters v x,v y,v z (or,, v ), but the same is true for since Hermiticity fixes Im 11 Im 0, Re 1 Re 1, Im 1 Im 1, an normalization Tr 1 gives us Re 1 Re 11. Hence Re 11, Re 1, an Im 1 are the only real egrees of freeom for a ensity operator on a two-imensional Hilbert space. Furthermore, Tr 1 4 Tr 1 v 1 4 Tr 1 v x x v y y v z z 1 1 v. Hence the length of the generalize Bloch vector is irectly relate to the purity of.
3 Although we first introuce the ensity operator as a means of representing uncertain preparations for an iniviual quantum system, it is equally well suite to escribing the average state of a collection of ientical systems. By ientical here we mean that each member of the ensemble lives in the same Hilbert space as every other, but the quantum states of the ifferent members of the ensemble will vary. So if we have a collection of twenty spin- 1 particles an prepare ten of them in the state z è an the other ten in x è, the ensemble ensity operator is 1 z èè z 1 x èè x. This type of ensemble ensity operator can be use to compute average values of observables such as S. In fact, the quantity M ân S correspons to the net magnetic moment (or magnetization) of a collection of N spin- 1 particles. When the magnetization vector has maximum length (here 0 â M â N Ò/), all the spins in the ensemble must be pointing in the same irection. When M 0, the irections of the spins in the ensemble are isotropically istribute. Hamiltonian ynamics for a ensity operator Assuming we have an ensemble representation for a given ensity operator, we can erive its equation of motion in the following way: p èè i Ò p è è è è p i Ò H è è è i Ò è H p H èè èè H i H,. Ò Since this erivation is clearly inepenent of the particular ensemble representation we choose for, it is vali in general. Similarly, we know that è T t,0 è exp ih t/ò è, è è T 0,t, so 3
4 Hence t p T t,0 èè T 0,t T t,0 p èè T 0,t T t,0 0 T 0,t. Tr t Tr T t,0 0 T 0,t T t,0 0 T 0,t Tr T t,0 0 T 0,t Tr 0, so Hamiltonian evolution preserves the length of the generalize Bloch vector. In fact, we can use the above methos to erive equations of motion for the Bloch vector itself: v i Tr i Tr i i Tr Ò i H, i Tr Ò i H, 1 v. Hence the evolutions of the three components will generally be couple. For Larmor precession in a static applie fiel, H S B an v i i Tr H, 1 v Ò 4 Tr i B 1 v i 1 v B 4 Tr ib x x ib y y ib z z 1 v x x v y y v z z 4 Tr 1 v x x v y y v z z B x x i B y y i B z z i 4 Tr x y B x v y v x B y x z B x v z v x B z y x B y v x v y B x 4 Tr y z B y v z v y B z z x B z v x v z B x, z y B z v y v z B y where the terms involving 1 have been roppe ue to cyclic property of the trace. Taking into account the orthogonality of the Pauli matrices, we are simply left with 4
5 v x 4 Tr x y z B y v z v y B z x z y B z v y v z B y 4 B yv z v y B z Tr x y z x z y B y v z v y B z, v y 4 B xv z v x B z Tr y x z y z x B z v x v z B x, v z 4 B xv y v x B y Tr z x y z y x B x v y v x B y. In vector notation, we thus recover v v B, S S B. We can also work with ensity operators in the rotating frame. Just as for state vectors we applie the transformation Ë è O z è, O z exp z t/, where is the angular frequency (about z) of the rotating frame, for ensity operators we can apply Ë O z O z an use with H Ë Ë i Ò H Ë, Ë O z HO z 1 iòo z O z 1 O z HO z 1 Ò z. Let s return to our holing fiel B 0 z plus rotating perturbation b 1 cos t x sin t y, H L S z B 0 b 1 cos t S x sin t S y, H 1 Ë Ò ' z b 1 x, where ' L. From this point on let s assume that we are always working in the rotating frame, so we can rop the primes (Ë). Then 5
6 v i i 4 Tr i ' z b 1 x, 1 v x x v y y v z z i 4 Tr i ' z b 1 x 1 v x x v y y v z z i 4 Tr i 1 v x x v y y v z z ' z b 1 x i 4 Tr i ' z v x x v y y b 1 x v y y v z z i 4 Tr i ' v x x v y y z b 1 v y y v z z x, so v x v y i 4 Tr x' z v y y x'v y y z ' v y, i 4 Tr y' z v x x y'v x x z i 4 Tr y b 1 x v z z y b 1 v z z x 'v x b 1v z 4 Tr y x z y z x 'v x b 1 v z, v z 4 i Tr z b 1 x v y y z b 1 v y y x Copying out the results, we have v x ' v y, v y 'v x b 1 v z, v z b 1 v y. b 1 v y 4 Tr z x y z y x b 1 v y. Given our previous results, we shoul not be surprise to iscover that these may also be written as the vector equation v v B eff, where B eff B 0 z b 1x. Don t forget that we have efine L B 0. Finally, we are reay to a relaxation (that is, amping an ephasing) terms! Phenomenologically, one can generally ientify two istinct relaxation timescales for an ensemble of spins in contact with a heat bath or reservoir. These are known as the longituinal relaxation time T 1 an the transverse relaxation time T. Roughly speaking, T 1 measures the time require for spins in the higher-energy eigenstate ( z è if 0 as we have been assuming in these notes) to ecay back own to the groun state (lower-energy eigenstate). Hence T 1 is associate with energy issipation, with longer T 1 implying a lower rate of energy loss into the environment. The transverse relaxation time T accounts for ephasing, by which we mean the tenency of environmental interactions to ranomly perturb the phase of Larmor precession. In the thir term of this course we will use concrete moels of the system-reservoir interaction to erive values for T 1 an T, but for now we will ust put them in by han. In terms of these two relaxation times, we can finally write own the full-blown Bloch Equations, 6
7 v x ' v y v x T 1, v y 'v x b 1 v z v y T 1, 0 v z b 1 v y v z v z T 1 1, where v 0 z is the value of v z at thermal equilibrium. We alreay unerstan that the Hamiltonian component of these equations (terms involving ' an b 1 ) simply escribe Larmor precession of the Bloch vector aroun the effective magnetic fiel in the rotating frame. In orer to get some feeling for the role of the issipative terms, let s look at the solutions with both ' an b 1 set to zero. That is, we maintain a holing fiel B 0 an work in a frame rotating at the Larmor frequency, but set the riving fiel to zero. The Bloch Equations become v x v x T 1, v y v y T 1, 0 v z v z v z T 1 1, for which we can easily write own the analytic solutions v x t v x 0 exp t/t, v y t v y 0 exp t/t, 0 v z t v z 0 v z exp t/t 1 v 0 z. Hence the components of the Bloch vector in the equatorial plane ecay exponentially to zero, with time constant T, an the z component of the Bloch vector ecays exponentially to its equilibrium value v 0 z with time constant T 1. A typical mechanism that woul contribute to T 1 for a collection of spins is collisional interactions. If our spins correspon to the nuclear spins of gas-phase atoms such as 19 Xe, then the spin of any given atom in the ensemble will be ranomly perturbe through collisions with other atoms in the gas. Since each atom carries a magnetic moment, any time two atoms pass close by one another they will each experience a time-epenent magnetic fiel. The envelope an irection of this fiel pulse will essentially be ranom, as it epens on the etails of the atomic spatial traectories uring the collision. If the collisions have a uration that is comparable to the inverse Larmor frequency (for whatever holing fiel is being applie), we may expect that the x an y components of the fiel pulse will have Fourier components near L an collision-inuce spin flips may occur. Appealing to general thermoynamic consierations, we may expect that the net effect of such spin flips will be to bring the energy istribution of the spins into equilibrium with the overall temperature of the gas (hence v 0 z ). Dephasing can occur if the system-reservoir interaction leas to ranomly varying B z.if the iniviual spins in the ensemble experience small perturbing fiels B in aition to the holing fiel B 0, then they will precess aroun z at slightly ifferent frequencies. Viewe in the rotating frame at the average Larmor frequency L, this means that the Bloch vectors corresponing to the members of the ensemble will graually fan out with respect to the longituinal angle. After some time, we may expect that the istribution in shoul become uniform, meaning that the ensemble-average Bloch vector woul have v x v y 0. The timescale T for such ephasing to occur can be erive explicitly for various moels of B. 7
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