DEGREE PROJECT IN TECHNOLOGY, FIRST CYCLE, 15 CREDITS STOCKHOLM, SWEDEN 2019 An Overview of Decoherence in the Context of Quantum Measurement with an Exactly Solvable Pure Decoherence Model FRODE BOMAN ANDRZEJ PERZANOWSKI KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES EXAMENSARBETE INOM TEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2019 En Översikt av Dekoherens i Kontexten av Kvantmätning med en Exakt Lösbar Ren Dekoherensmodell FRODE BOMAN ANDRZEJ PERZANOWSKI KTH SKOLAN FÖR TEKNIKVETENSKAP An Overview of Decoherence in the Context of Quantum Measurement with an Exactly Solvable Pure Decoherence Model Frode Boman [email protected] Andrzej Perzanowski [email protected] Supervisor: Edwin Langmann Abstract We explain the concept of decoherence by first providing basic back- ground information and then working out a simple model intended to represent a Stern-Gerlach experiment. The model describes a spin- 1/2 quantum system and a device consisting of a quantum particle. Using Lindblad dynamics, the model is solved, revealing the effects of decoherence on the system. Specifically, we show that decoherence is sufficient for driving the off-diagonal of the system-device density operator to zero, providing a possible explanation to the fact of wave function collapse. Furthermore, comparisons are made with related work on a similar, exact model which includes relaxation. Sammanfattning Vi f¨orklarar konceptet dekoherens genom att f¨orst ge bakgrundsin- formation och sedan l¨osa en enkel modell som ¨ar menad att represen- tera ett Stern-Gerlach experiment. Modellen ¨ar ett spin-1/2 kvantsy- stem och en m¨atapparat som best˚arav en kvantpartikel. Genom att anv¨anda Lindbladdynamik kan vi l¨osa modellen och visa effekterna som dekoherens har p˚asystemet. Specifikt visar vi att dekoherens ¨ar tillr¨ackligt f¨or att driva de icke-diagonala elementen av system-apparat densitetsoperatorn till noll, vilket ger en m¨ojlig f¨orklaring till faktumet av v˚agfunktionskollaps. Vidare g¨ors j¨amf¨orelser med relaterad forsk- ning av en liknande, exakt modell som inkluderar relaxation. 1 Contents 1 Background 3 1.1 The statistical interpretation . 3 1.2 Measurement and the measurement problem . 4 1.3 Density matrices . 6 1.4 Time evolution . 7 1.5 Environment and decoherence . 8 1.6 Master equations . 10 2 Method 12 2.1 Overview . 12 2.2 The model . 13 2.3 The master equation . 14 2.4 Characteristic times for decoherence and relaxation . 15 3 Results 16 4 Discussion 17 5 Conclusions 26 6 References 28 A Appendix 30 A.1 Spin diagonal matrix element . 30 A.2 Spin off-diagonal matrix element . 31 A.3 Solutions in momentum representation . 34 2 1 Background 1.1 The statistical interpretation Since the inception of quantum mechanics it has been regarded, both by scholars and laypersons, as a probabilistic theory of nature. This is despite the fact that the Schr¨odingerequation, which wholly describes the time evolu- tion of a quantum stystem, is entirely deterministic. Indeed, the probabilistic nature of quantum mechanics appears not as a result of its governing equa- tions but instead as a result of something much less tactile: the measurement process. The person most often credited with introducing probability into the the- ory of quantum mechanics is Max Born, who in the year 1926 formulated what has since been referred to as the Born rule (Born 1926). Since its ac- ceptance, the Born rule has shaped how we interpret quantum mechanics, and indeed the Universe (Landsman 2009). Still, the theory which leads one to the Born rule is unsatisfactory. Max Born, in his 1926 paper, did not in fact derive the rule. Instead, he concluded that is must be true, as there was no other way to interpret the results of his calculations. The lack of founda- tion for the rule led to dispute regarding how it should be interpreted, and multiple major interpretations of quantum mechanics have been formulated to make sense of it. In order to formulate the problem, known today as the measurement prob- lem, we introduce the mathematical language most suited for such discussions (Griffiths & Schroeter 2018, p. 91-130): In quantum mechanics, wave func- tions (i.e., solutions to the Schr¨odingerequation) are described as quantum states. The state vector representation of the wave function (x; t) is given by its ket; a vector in Hilbert space H. The measureable properties of the system (position, momentum, spin, etc.) are linear operators on H called observables. The possible states of the system are the eigenvalues of the corresponding observable. Due to this, it is common to describe a state j i by its projection onto the orthonormal eigenspace of the observable. For non-degenerate discrete spectra this means writing X j i = hφkj i jφki: (1) k 3 Here, jφi is the eigenvector, or eigenstate, of the observable. Examples of observables with discrete spectra are spin and angular momentum. For con- tinuous spectra such as position and momentum, we instead have Z j i = hφkj i jφki dk: (2) When the observable O^ operates on the system we thus get (in the discrete case) ^ X ^ X O j i = hφkj i O jφki = hφkj i φk jφki (3) k k where φk is the eigenvalue that corresponds to the eigenstate jφki. This is known as as superposition. Equation (3) leads to the idea that the state of the system is a sum of all possible states. This is not observed when mea- surements are made, nor is it clear what such a measurement result would even look like. To resolve this issue we now formulate the Born rule: If one makes a measurement of an observable, then as a result one will get a single eigenvalue of O^. The probability that one will measure a specific 2 eigenvalue φk is given by j hφkj i j . For the continuous case, the analogous rule is that a state will be found in the range dk about φk with a probability 2 j hφkj i j dk. For the purposes of this exposition, we need not introduce the case of degenerate spectra. The transition from superposition to a state which is well-defined is referred to as wave function collapse. It is this process which has given rise to the measurement problem and to a whole area of research: quantum measure- ment theory. 1.2 Measurement and the measurement problem Early quantum measurement theory was notably contributed to by John von Neumann, who in his book Mathematische Grundlagen der Quanten- mechanik (Mathematical Foundations of Quantum Mechanics) devoted a chapter to the subject. Among other ideas, von Neumann modelled the measurement apparatus, some device D (Here, D is the Hilbert space of the device. In some literature, this may be denoted HD), as a quantum system (von Neumann 2018). During the measurement process, the Hilbert space 4 of interest is the tensor product space S ⊗ D (Allahverdyan et al. 2013). A simple measurement model looks like the following: Let a system S have N possible states fjsiigi=1. If we wish to measure which state the system is in then our device should take the value jdii if the system is in state jsii. As an example, lets say that the system is in a state jski. Then our device, initially in a ready-state jd0i, wil, via a measurement, transition into the state jdki. Observers can then infer that the system is in the state jski, since they can see that the measurement device is in the state jdki. For a general system, P in a superposition of states jsi = i ci jsii, this process becomes X X ( ci jsii) ⊗ jd0i ! ci jsii ⊗ jdii: (4) i i Were the measurement device macroscopic, reading the measurement value from its pointer would be no issue. This is problematic because as the sys- tem is in a superposition of states, so too is the measurement device. Clearly, wave function collapse is yet to take place. A possible (though possibly un- satisfactory) explanation for this may be that it is unreasonable to describe a macroscopic system as a quantum one. The idea that the macroscopic is inherently different from the microscopic runs into difficult questions, such as there not being a clear distinguishing line between them. Regardless, we can consider the case where the apparatus is not macroscopic. Then, equa- tion (4) should hold. This is referred to as a pre-measurement. It is called that because it is not a true measurement, in that no information can be gathered from the microscopic device: It has no pointer for an observer to note down its state. Still, a measurement has technically taken place, in that the system has interacted with a device. The information regarding the state of the system could now be obtained by performing a measurement on the device. It may seem like nothing has been accomplished by performing the pre-measurement; that we have merely moved the problem from the sys- tem to the device. As far as the measurement problem is concerned, this is true, although practically it may be useful to convert sensitive information into more stable information (this is often referred to as preparation of a state). An interesting note is that it was this line of reasoning which led von Neumann to adhere to the controversial idea that wave function collapse is caused by consciousness (an idea which is now referred to as the Neumann- Wigner interpretation). As one introduced more and more devices, one is merely moving the problem along, until eventually, the mind of the observer performs the final measurement. This idea is mostly rejected among modern 5 physicists (Schlosshauer et al. 2013).